\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/input schaum2.input}
\author{Timothy Daly}
\maketitle
\eject
\tableofcontents
\eject
\section{\cite{1}:14.84~~~~~$\displaystyle
\int{\frac{dx}{\sqrt{ax+b}}}$}
$$\int{\frac{1}{\sqrt{ax+b}}}=
\frac{2\sqrt{ax+b}}{a}
$$
<<*>>=
)spool schaum2.output
)set message test on
)set message auto off
)clear all

--S 1
aa:=integrate(1/sqrt(a*x+b),x)
--R 
--R
--R          +-------+
--R        2\|a x + b
--R   (1)  -----------
--R             a
--R                                          Type: Union(Expression Integer,...)
--E 

--S 2
bb:=(2*sqrt(a*x+b))/a
--R 
--R
--R          +-------+
--R        2\|a x + b
--R   (2)  -----------
--R             a
--R                                                     Type: Expression Integer
--E

--S 3      14:84 Schaums and Axiom agree
cc:=aa-bb
--R 
--R
--R   (3)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.85~~~~~$\displaystyle
\int{\frac{x~dx}{\sqrt{ax+b}}}$}
$$\int{\frac{x}{\sqrt{ax+b}}}=
\frac{2(ax-2b)}{3a^2}\sqrt{ax+b}
$$
<<*>>=
)clear all

--S 4
aa:=integrate(x/sqrt(a*x+b),x)
--R 
--R
--R                    +-------+
--R        (2a x - 4b)\|a x + b
--R   (1)  ---------------------
--R                   2
--R                 3a
--R                                          Type: Union(Expression Integer,...)
--E 

--S 5
bb:=(2*(a*x-2*b))/(3*a^2)*sqrt(a*x+b)
--R 
--R
--R                    +-------+
--R        (2a x - 4b)\|a x + b
--R   (2)  ---------------------
--R                   2
--R                 3a
--R                                                     Type: Expression Integer
--E

--S 6      14:85 Schaums and Axiom agree
cc:=aa-bb
--R 
--R
--R   (3)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.86~~~~~$\displaystyle
\int{\frac{x^2~dx}{\sqrt{ax+b}}}$}
$$\int{\frac{x}{\sqrt{ax+b}}}=
\frac{2(3a^2x^2-4abx+8b^2)}{15a^2}\sqrt{ax+b}
$$
<<*>>=
)clear all

--S 7
aa:=integrate(x^2/sqrt(a*x+b),x)
--R 
--R
--R           2 2               2  +-------+
--R        (6a x  - 8a b x + 16b )\|a x + b
--R   (1)  ---------------------------------
--R                          3
--R                       15a
--R                                          Type: Union(Expression Integer,...)
--E 

--S 8
bb:=(2*(3*a^2*x^2-4*a*b*x+8*b^2))/(15*a^3)*sqrt(a*x+b)
--R 
--R
--R           2 2               2  +-------+
--R        (6a x  - 8a b x + 16b )\|a x + b
--R   (2)  ---------------------------------
--R                          3
--R                       15a
--R                                                     Type: Expression Integer
--E

--S 9      14:86 Schaums and Axiom agree
cc:=aa-bb
--R 
--R
--R   (3)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.87~~~~~$\displaystyle
\int{\frac{dx}{x\sqrt{ax+b}}}$}
$$\int{\frac{1}{x\sqrt{ax+b}}}=
\left\{
\begin{array}{l}
\displaystyle
\frac{1}{\sqrt{b}}~\ln
\left(\frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}}\right)\\
\displaystyle
\frac{2}{\sqrt{-b}}~\tan^{-1}\sqrt{\frac{ax+b}{-b}}
\end{array}
\right.
$$

Note: the first answer assumes $b > 0$ and the second assumes $b < 0$.
<<*>>=
)clear all

--S 10
aa:=integrate(1/(x*sqrt(a*x+b)),x)
--R 
--R
--R                  +-------+              +-+           +---+ +-------+
--R             - 2b\|a x + b  + (a x + 2b)\|b           \|- b \|a x + b
--R         log(-------------------------------)   2atan(----------------)
--R                            x                                 b
--R   (1)  [------------------------------------,- -----------------------]
--R                          +-+                             +---+
--R                         \|b                             \|- b
--R                                     Type: Union(List Expression Integer,...)
--E 
@
Cleary Spiegel's first answer assumes $b > 0$:
<<*>>=
--S 11
bb1:=1/sqrt(b)*log((sqrt(a*x+b)-sqrt(b))/(sqrt(a*x+b)+sqrt(b)))
--R 
--R
--R             +-------+    +-+
--R            \|a x + b  - \|b
--R        log(-----------------)
--R             +-------+    +-+
--R            \|a x + b  + \|b
--R   (2)  ----------------------
--R                  +-+
--R                 \|b
--R                                                     Type: Expression Integer
--E
@
So we try the difference of the two results
<<*>>=
--S 12
cc11:=aa.1-bb1
--R
--R               +-------+    +-+             +-------+              +-+
--R              \|a x + b  - \|b         - 2b\|a x + b  + (a x + 2b)\|b
--R        - log(-----------------) + log(-------------------------------)
--R               +-------+    +-+                       x
--R              \|a x + b  + \|b
--R   (3)  ---------------------------------------------------------------
--R                                       +-+
--R                                      \|b
--R                                                     Type: Expression Integer
--E
@
But the results don't simplify to 0. So we try some other tricks.

Since both functions are of the form log(f(x))/sqrt(b) we extract
the f(x) from each. First we get the function from Axiom's first answer:
<<*>>=
--S 13
ff:=exp(aa.1*sqrt(b))
--R
--R             +-------+              +-+
--R        - 2b\|a x + b  + (a x + 2b)\|b
--R   (4)  -------------------------------
--R                       x
--R                                                     Type: Expression Integer
--E
@
and we get the same form from Spiegel's answer
<<*>>=
--S 14
gg:=exp(bb1*sqrt(b))
--R
--R         +-------+    +-+
--R        \|a x + b  - \|b
--R   (5)  -----------------
--R         +-------+    +-+
--R        \|a x + b  + \|b
--R                                                     Type: Expression Integer
--E
@
We can change Spiegel's form into Axiom's form because they differ by
the constant a*sqrt(b). To see this we multiply the numerator and
denominator by $1 == (sqrt(a*x+b) - sqrt(b))/(sqrt(a*x+b) - sqrt(b))$.

First we multiply the numerator by $(sqrt(a*x+b) - sqrt(b))$
<<*>>=
--S 15
gg1:=gg*(sqrt(a*x+b) - sqrt(b))
--R
--R            +-+ +-------+
--R        - 2\|b \|a x + b  + a x + 2b
--R   (6)  ----------------------------
--R               +-------+    +-+
--R              \|a x + b  + \|b
--R                                                     Type: Expression Integer
--E
@
Now we multiply the denominator by $(sqrt(a*x+b) - sqrt(b))$
<<*>>=
--S 16
gg2:=gg1/(sqrt(a*x+b) - sqrt(b))
--R
--R            +-+ +-------+
--R        - 2\|b \|a x + b  + a x + 2b
--R   (7)  ----------------------------
--R                     a x
--R                                                     Type: Expression Integer
--E
@
and now we multiply by the integration constant $a*sqrt(b)$
<<*>>=
--S 17
gg3:=gg2*(a*sqrt(b))
--R
--R             +-------+              +-+
--R        - 2b\|a x + b  + (a x + 2b)\|b
--R   (8)  -------------------------------
--R                       x
--R                                                     Type: Expression Integer
--E
@
and when we difference this with ff, the Axiom answer we get:
<<*>>=
--S 18     14:87a Schaums and Axiom differ by a constant
ff-gg3
--R
--R   (9)  0
--R                                                     Type: Expression Integer
--E
@
So the constant of integration difference is $a*sqrt(b)$

Now we look at the second equations. We difference Axiom's second answer
from Spiegel's answer:
<<*>>=
--S 19
t1:=aa.2-bb1
--R
--R                      +-------+    +-+               +---+ +-------+
--R            +---+    \|a x + b  - \|b       +-+     \|- b \|a x + b
--R         - \|- b log(-----------------) - 2\|b atan(----------------)
--R                      +-------+    +-+                      b
--R                     \|a x + b  + \|b
--R   (10)  ------------------------------------------------------------
--R                                   +---+ +-+
--R                                  \|- b \|b
--R                                                     Type: Expression Integer
--E
@
and again they do not simplify to zero. But we can show that both answers
differ by a constant because the derivative is zero:
<<*>>=
--S 20
D(t1,x)
--R
--R   (11)  0
--R                                                     Type: Expression Integer
--E
@

Rather than find the constant this time we will differentiate both
answers and compare them with the original equation.
<<*>>=
--S 21
target:=1/(x*sqrt(a*x+b))
--R
--R              1
--R   (12)  -----------
--R           +-------+
--R         x\|a x + b
--R                                                     Type: Expression Integer
--E
@
and we select the second Axiom solution
<<*>>=
--S 22
aa2:=aa.2
--R
--R                  +---+ +-------+
--R                 \|- b \|a x + b
--R           2atan(----------------)
--R                         b
--R   (13)  - -----------------------
--R                     +---+
--R                    \|- b
--R                                                     Type: Expression Integer
--E
@
take its derivative
<<*>>=
--S 23
ad2:=D(aa2,x)
--R
--R              1
--R   (14)  -----------
--R           +-------+
--R         x\|a x + b
--R                                                     Type: Expression Integer
--E
@
When we take the difference of Axiom's input and the derivative of the
output we see:
<<*>>=
--S 24
ad2-target
--R
--R   (15)  0
--R                                                     Type: Expression Integer
--E
@
Thus the original equation and Axiom's derivative of the integral are equal.

Now we do the same with Spiegel's answer. We take the derivative of his
answer.
<<*>>=
--S 25
ab1:=D(bb1,x)
--R
--R                +-------+    +-+
--R               \|a x + b  + \|b
--R   (16)  ----------------------------
--R           +-+ +-------+      2
--R         x\|b \|a x + b  + a x  + b x
--R                                                     Type: Expression Integer
--E
@
and we difference it from the original equation
<<*>>=
--S 26     14:87b Schaums and Axiom differ by a constant
ab1-target
--R
--R   (17)  0
--R                                                     Type: Expression Integer
--E
@
Thus the original equation and Spiegel's derivative of the integral are equal.

So we can conclude that both second answers are correct although they differ
by a constant of integration.

\section{\cite{1}:14.88~~~~~$\displaystyle
\int{\frac{dx}{x^2\sqrt{ax+b}}}$}
$$\int{\frac{1}{x^2\sqrt{ax+b}}}=
-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}~\int{\frac{1}{x\sqrt{ax+b}}}
$$
<<*>>=
)clear all

--S 27
aa:=integrate(1/(x^2*sqrt(a*x+b)),x)
--R 
--R
--R   (1)
--R               +-------+              +-+
--R            2b\|a x + b  + (a x + 2b)\|b       +-+ +-------+
--R    a x log(-----------------------------) - 2\|b \|a x + b
--R                          x
--R   [--------------------------------------------------------,
--R                                 +-+
--R                            2b x\|b
--R              +---+ +-------+
--R             \|- b \|a x + b      +---+ +-------+
--R    a x atan(----------------) - \|- b \|a x + b
--R                     b
--R    ---------------------------------------------]
--R                          +---+
--R                      b x\|- b
--R                                     Type: Union(List Expression Integer,...)
--E 
@

In order to write down the book answer we need to first take the
integral which has two results
<<*>>=
--S 28
dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R 
--R
--R                  +-------+              +-+           +---+ +-------+
--R             - 2b\|a x + b  + (a x + 2b)\|b           \|- b \|a x + b
--R         log(-------------------------------)   2atan(----------------)
--R                            x                                 b
--R   (2)  [------------------------------------,- -----------------------]
--R                          +-+                             +---+
--R                         \|b                             \|- b
--R                                     Type: Union(List Expression Integer,...)
--E
@
and derive two results for the book answer. The first result assumes
$b > 0$
<<*>>=
--S 29
bb1:=-sqrt(a*x+b)/(b*x)-a/(2*b)*dd.1
--R 
--R
--R                       +-------+              +-+
--R                  - 2b\|a x + b  + (a x + 2b)\|b       +-+ +-------+
--R        - a x log(-------------------------------) - 2\|b \|a x + b
--R                                 x
--R   (3)  ------------------------------------------------------------
--R                                       +-+
--R                                  2b x\|b
--R                                                     Type: Expression Integer
--E
@
and the second result assumes $b < 0$.
<<*>>=
--S 30
bb2:=-sqrt(a*x+b)/(b*x)-a/(2*b)*dd.2
--R 
--R
--R                  +---+ +-------+
--R                 \|- b \|a x + b      +---+ +-------+
--R        a x atan(----------------) - \|- b \|a x + b
--R                         b
--R   (4)  ---------------------------------------------
--R                              +---+
--R                          b x\|- b
--R                                                     Type: Expression Integer
--E
@

So we compute the difference of Axiom's first result with Spiegel's
first result
<<*>>=
--S 31
cc11:=bb1-aa.1
--R
--R   (5)
--R                  +-------+              +-+
--R               2b\|a x + b  + (a x + 2b)\|b
--R       - a log(-----------------------------)
--R                             x
--R     + 
--R                    +-------+              +-+
--R               - 2b\|a x + b  + (a x + 2b)\|b
--R       - a log(-------------------------------)
--R                              x
--R  /
--R        +-+
--R     2b\|b
--R                                                     Type: Expression Integer
--E
@
we compute its derivative
<<*>>=
--S 32
D(cc11,x)
--R
--R   (6)  0
--R                                                     Type: Expression Integer
--E
@
and we can see that the answers differ by a constant, the constant of
integration. So Axiom's first answer should differentiate back to the target
equation.
<<*>>=
--S 33
target:=1/(x^2*sqrt(a*x+b))
--R
--R              1
--R   (7)  ------------
--R         2 +-------+
--R        x \|a x + b
--R                                                     Type: Expression Integer
--E
@
We differentiate Axiom's first answer
<<*>>=
--S 34
ad1:=D(aa.1,x)
--R
--R                             +-+ +-------+              2
--R                  (a x + 2b)\|b \|a x + b  + 2a b x + 2b
--R   (8)  ----------------------------------------------------------
--R               3     2 2  +-------+     2 4         3     2 2  +-+
--R        (2a b x  + 2b x )\|a x + b  + (a x  + 3a b x  + 2b x )\|b
--R                                                     Type: Expression Integer
--E
@
and subtract it from the target equation
<<*>>=
--S 35
ad1-target
--R
--R   (9)  0
--R                                                     Type: Expression Integer
--E
@
and now we do the same with first Spiegel's answer:
<<*>>=
--S 36
bd1:=D(bb1,x)
--R
--R                                +-+ +-------+              2
--R                   (- a x - 2b)\|b \|a x + b  + 2a b x + 2b
--R   (10)  ------------------------------------------------------------
--R                3     2 2  +-------+       2 4         3     2 2  +-+
--R         (2a b x  + 2b x )\|a x + b  + (- a x  - 3a b x  - 2b x )\|b
--R                                                     Type: Expression Integer
--E
@
and we subtract it from the target
<<*>>=
--S 37     
bd1-target
--R
--R   (11)  0
--R                                                     Type: Expression Integer
--E
@
so we know that the two first answers are both correct and that their
integrals differ by a constant.

Now we look at the second answers. We difference the answers and can
see immediately that they are equal.
<<*>>=
--S 38     14:88 Schaums and Axiom differ by a constant
cc22:=bb2-aa.2
--R 
--R
--R   (12)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.89~~~~~$\displaystyle
\int{\sqrt{ax+b}~dx}$}
$$\int{\sqrt{ax+b}}=
\frac{2\sqrt{(ax+b)^3}}{3a}
$$
<<*>>=
)clear all

--S 39
aa:=integrate(sqrt(a*x+b),x)
--R 
--R
--R                    +-------+
--R        (2a x + 2b)\|a x + b
--R   (1)  ---------------------
--R                  3a
--R                                          Type: Union(Expression Integer,...)
--E 
@
<<*>>=
--S 40
bb:=(2*sqrt((a*x+b)^3))/(3*a)
--R 
--R
--R          +----------------------------+
--R          | 3 3     2   2       2     3
--R        2\|a x  + 3a b x  + 3a b x + b
--R   (2)  --------------------------------
--R                       3a
--R                                                     Type: Expression Integer
--E
@
<<*>>=
--S 41
cc:=aa-bb
--R
--R            +----------------------------+
--R            | 3 3     2   2       2     3                +-------+
--R        - 2\|a x  + 3a b x  + 3a b x + b   + (2a x + 2b)\|a x + b
--R   (3)  ----------------------------------------------------------
--R                                    3a
--R                                                     Type: Expression Integer
--E
@
Since this didn't simplify we could check each answer using the derivative
<<*>>=
--S 42
target:=sqrt(a*x+b)
--R
--R         +-------+
--R   (4)  \|a x + b
--R                                                     Type: Expression Integer
--E
@
We take the derivative of Axiom's answer
<<*>>=
--S 43
t1:=D(aa,x)
--R
--R          a x + b
--R   (5)  ----------
--R         +-------+
--R        \|a x + b
--R                                                     Type: Expression Integer
--E
@
And we subtract the target from the derivative of Axiom's answer
<<*>>=
--S 44
t1-target
--R
--R   (6)  0
--R                                                     Type: Expression Integer
--E
@
So they are equal. Now we do the same with Spiegel's answer
<<*>>=
--S 45
t2:=D(bb,x)
--R
--R                2 2             2
--R               a x  + 2a b x + b
--R   (7)  -------------------------------
--R         +----------------------------+
--R         | 3 3     2   2       2     3
--R        \|a x  + 3a b x  + 3a b x + b
--R                                                     Type: Expression Integer
--E
@
The numerator is
<<*>>=
--S 46
nn:=(a*x+b)^2
--R
--R         2 2             2
--R   (8)  a x  + 2a b x + b
--R                                                     Type: Polynomial Integer
--E
@
<<*>>=
--S 47
mm:=(a*x+b)^3
--R
--R         3 3     2   2       2     3
--R   (9)  a x  + 3a b x  + 3a b x + b
--R                                                     Type: Polynomial Integer
--E
@
which expands to Spiegel's version.
<<*>>=
--S 48     14:89 Schaums and Axiom differ by a constant
result=nn/sqrt(mm)
--R
--R                         2 2             2
--R                        a x  + 2a b x + b
--R   (10)  result= -------------------------------
--R                  +----------------------------+
--R                  | 3 3     2   2       2     3
--R                 \|a x  + 3a b x  + 3a b x + b
--R                                            Type: Equation Expression Integer
--E
@
and this reduces to $\sqrt{ax+b}$

\section{\cite{1}:14.90~~~~~$\displaystyle
\int{x\sqrt{ax+b}~dx}$}
$$\int{x\sqrt{ax+b}}=
\frac{2(3ax-2b)}{15a^2}~\sqrt{(ax+b)^3}
$$
<<*>>=
)clear all

--S 49
aa:=integrate(x*sqrt(a*x+b),x)
--R 
--R
--R           2 2              2  +-------+
--R        (6a x  + 2a b x - 4b )\|a x + b
--R   (1)  --------------------------------
--R                         2
--R                      15a
--R                                          Type: Union(Expression Integer,...)
--E 

--S 50
bb:=(2*(3*a*x-2*b))/(15*a^2)*sqrt((a*x+b)^3)
--R 
--R
--R                    +----------------------------+
--R                    | 3 3     2   2       2     3
--R        (6a x - 4b)\|a x  + 3a b x  + 3a b x + b
--R   (2)  ------------------------------------------
--R                              2
--R                           15a
--R                                                     Type: Expression Integer
--E

--S 51
cc:=aa-bb
--R
--R   (3)
--R                     +----------------------------+
--R                     | 3 3     2   2       2     3
--R       (- 6a x + 4b)\|a x  + 3a b x  + 3a b x + b
--R     + 
--R          2 2              2  +-------+
--R       (6a x  + 2a b x - 4b )\|a x + b
--R  /
--R        2
--R     15a
--R                                                     Type: Expression Integer
--E

--S 52     14:90 Schaums and Axiom agree
dd:=rootSimp cc
--R
--R   (4)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.91~~~~~$\displaystyle
\int{x^2\sqrt{ax+b}~dx}$}
$$\int{x^2\sqrt{ax+b}}=
\frac{2(15a^2x^2-12abx+8b^2)}{105a^2}~\sqrt{(a+bx)^3}
$$
Note: the sqrt term is almost certainly $\sqrt{(ax+b)}$
<<*>>=
)clear all

--S 53
aa:=integrate(x^2*sqrt(a*x+b),x)
--R 
--R
--R            3 3     2   2       2       3  +-------+
--R        (30a x  + 6a b x  - 8a b x + 16b )\|a x + b
--R   (1)  --------------------------------------------
--R                                3
--R                            105a
--R                                          Type: Union(Expression Integer,...)
--E 

--S 54
bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^3)*sqrt((a+b*x)^3)
--R 
--R
--R                                  +----------------------------+
--R            2 2                2  | 3 3       2 2     2       3
--R        (30a x  - 24a b x + 16b )\|b x  + 3a b x  + 3a b x + a
--R   (2)  --------------------------------------------------------
--R                                      3
--R                                  105a
--R                                                     Type: Expression Integer
--E

--S 55     14:91 Axiom cannot simplify this expression. Schaums typo?
cc:=aa-bb
--R
--R   (3)
--R                                   +----------------------------+
--R             2 2                2  | 3 3       2 2     2       3
--R       (- 30a x  + 24a b x - 16b )\|b x  + 3a b x  + 3a b x + a
--R     + 
--R           3 3     2   2       2       3  +-------+
--R       (30a x  + 6a b x  - 8a b x + 16b )\|a x + b
--R  /
--R         3
--R     105a
--R                                                     Type: Expression Integer
--E

@
Notice that if we factor the numerator of 'aa' we get an expression that
differs from schaums on by the order of the variables in the square root.
(We can square the term (a*x+b) and drag it under the square root to get
the cubic term). It appears that Schaums has a typo.
<<*>>=
--S 56
factor numer aa
--R
--R                      2 2               2  +-------+
--R   (4)  2(a x + b)(15a x  - 12a b x + 8b )\|a x + b
--RType: Factored SparseMultivariatePolynomial(Integer,Kernel Expression Integer)
--E
@

\section{\cite{1}:14.92~~~~~$\displaystyle
\int{\frac{\sqrt{ax+b}}{x}~dx}$}
$$\int{\frac{\sqrt{ax+b}}{x}}=
2\sqrt{ax+b}+b~\int{\frac{1}{x\sqrt{ax+b}}}
$$
<<*>>=
)clear all

--S 57
aa:=integrate(sqrt(a*x+b)/x,x)
--R 
--R
--R   (1)
--R                +-+ +-------+
--R     +-+    - 2\|b \|a x + b  + a x + 2b      +-------+
--R   [\|b log(----------------------------) + 2\|a x + b ,
--R                          x
--R                   +-------+
--R        +---+     \|a x + b       +-------+
--R    - 2\|- b atan(----------) + 2\|a x + b ]
--R                     +---+
--R                    \|- b
--R                                     Type: Union(List Expression Integer,...)
--E 

--S 58
dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R 
--R
--R                  +-------+              +-+           +---+ +-------+
--R             - 2b\|a x + b  + (a x + 2b)\|b           \|- b \|a x + b
--R         log(-------------------------------)   2atan(----------------)
--R                            x                                 b
--R   (2)  [------------------------------------,- -----------------------]
--R                          +-+                             +---+
--R                         \|b                             \|- b
--R                                     Type: Union(List Expression Integer,...)
--E

--S 59
bb1:=2*sqrt(a*x+b)+b*dd.1
--R 
--R
--R                   +-------+              +-+
--R              - 2b\|a x + b  + (a x + 2b)\|b       +-+ +-------+
--R        b log(-------------------------------) + 2\|b \|a x + b
--R                             x
--R   (3)  --------------------------------------------------------
--R                                   +-+
--R                                  \|b
--R                                                     Type: Expression Integer
--E

--S 60
bb2:=2*sqrt(a*x+b)+b*dd.2
--R 
--R
--R                   +---+ +-------+
--R                  \|- b \|a x + b       +---+ +-------+
--R        - 2b atan(----------------) + 2\|- b \|a x + b
--R                          b
--R   (4)  -----------------------------------------------
--R                              +---+
--R                             \|- b
--R                                                     Type: Expression Integer
--E

--S 61
cc11:=bb1-aa.1
--R 
--R
--R   (5)
--R              +-------+              +-+              +-+ +-------+
--R         - 2b\|a x + b  + (a x + 2b)\|b           - 2\|b \|a x + b  + a x + 2b
--R   b log(-------------------------------) - b log(----------------------------)
--R                        x                                       x
--R   ----------------------------------------------------------------------------
--R                                        +-+
--R                                       \|b
--R                                                     Type: Expression Integer
--E

--S 62
cc12:=bb1-aa.2
--R 
--R
--R                   +-------+              +-+                     +-------+
--R              - 2b\|a x + b  + (a x + 2b)\|b       +---+ +-+     \|a x + b
--R        b log(-------------------------------) + 2\|- b \|b atan(----------)
--R                             x                                      +---+
--R                                                                   \|- b
--R   (6)  --------------------------------------------------------------------
--R                                         +-+
--R                                        \|b
--R                                                     Type: Expression Integer
--E

--S 63
cc21:=bb2-aa.1
--R 
--R
--R   (7)
--R                       +-+ +-------+                        +---+ +-------+
--R      +---+ +-+    - 2\|b \|a x + b  + a x + 2b            \|- b \|a x + b
--R   - \|- b \|b log(----------------------------) - 2b atan(----------------)
--R                                 x                                 b
--R   -------------------------------------------------------------------------
--R                                      +---+
--R                                     \|- b
--R                                                     Type: Expression Integer
--E

--S 64
cc22:=bb2-aa.2
--R 
--R
--R                   +---+ +-------+             +-------+
--R                  \|- b \|a x + b             \|a x + b
--R        - 2b atan(----------------) - 2b atan(----------)
--R                          b                      +---+
--R                                                \|- b
--R   (8)  -------------------------------------------------
--R                               +---+
--R                              \|- b
--R                                                     Type: Expression Integer
--E

--S 65     14:92 Schaums and Axiom agree
dd22:=ratDenom cc22
--R
--R   (9)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.93~~~~~$\displaystyle
\int{\frac{\sqrt{ax+b}}{x^2}~dx}$}
$$\int{\frac{\sqrt{ax+b}}{x^2}}=
-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}~\int{\frac{1}{x\sqrt{ax+b}}}
$$
<<*>>=
)clear all

--S 65
aa:=integrate(sqrt(a*x+b)/x^2,x)
--R 
--R
--R   (1)
--R                 +-------+              +-+
--R            - 2b\|a x + b  + (a x + 2b)\|b       +-+ +-------+
--R    a x log(-------------------------------) - 2\|b \|a x + b
--R                           x
--R   [----------------------------------------------------------,
--R                                 +-+
--R                              2x\|b
--R                +---+ +-------+
--R               \|- b \|a x + b      +---+ +-------+
--R    - a x atan(----------------) - \|- b \|a x + b
--R                       b
--R    -----------------------------------------------]
--R                          +---+
--R                        x\|- b
--R                                     Type: Union(List Expression Integer,...)
--E 

--S 66
dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R 
--R
--R                  +-------+              +-+           +---+ +-------+
--R             - 2b\|a x + b  + (a x + 2b)\|b           \|- b \|a x + b
--R         log(-------------------------------)   2atan(----------------)
--R                            x                                 b
--R   (2)  [------------------------------------,- -----------------------]
--R                          +-+                             +---+
--R                         \|b                             \|- b
--R                                     Type: Union(List Expression Integer,...)
--E

--S 67
bb1:=-sqrt(a*x+b)/x+a/2*dd.1
--R 
--R
--R                     +-------+              +-+
--R                - 2b\|a x + b  + (a x + 2b)\|b       +-+ +-------+
--R        a x log(-------------------------------) - 2\|b \|a x + b
--R                               x
--R   (3)  ----------------------------------------------------------
--R                                     +-+
--R                                  2x\|b
--R                                                     Type: Expression Integer
--E

--S 68
bb2:=-sqrt(a*x+b)/x+a/2*dd.2
--R 
--R
--R                    +---+ +-------+
--R                   \|- b \|a x + b      +---+ +-------+
--R        - a x atan(----------------) - \|- b \|a x + b
--R                           b
--R   (4)  -----------------------------------------------
--R                              +---+
--R                            x\|- b
--R                                                     Type: Expression Integer
--E

--S 69
cc11:=bb1-aa.1
--R 
--R
--R   (5)  0
--R                                                     Type: Expression Integer
--E

--S 70
cc21:=bb-aa.1
--R 
--R
--R   (6)
--R                  +-------+              +-+
--R             - 2b\|a x + b  + (a x + 2b)\|b       +-+ +-------+         +-+
--R   - a x log(-------------------------------) + 2\|b \|a x + b  + 2bb x\|b
--R                            x
--R   ------------------------------------------------------------------------
--R                                       +-+
--R                                    2x\|b
--R                                                     Type: Expression Integer
--E

--S 71
cc12:=bb1-aa.2
--R 
--R
--R   (7)
--R                   +-------+              +-+                +---+ +-------+
--R     +---+    - 2b\|a x + b  + (a x + 2b)\|b        +-+     \|- b \|a x + b
--R   a\|- b log(-------------------------------) + 2a\|b atan(----------------)
--R                             x                                      b
--R   --------------------------------------------------------------------------
--R                                     +---+ +-+
--R                                   2\|- b \|b
--R                                                     Type: Expression Integer
--E

--S 72     14:93 Schaums and Axiom agree
cc22:=bb2-aa.2
--R 
--R
--R   (8)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.94~~~~~$\displaystyle
\int{\frac{x^m}{\sqrt{ax+b}}~dx}$}
$$\int{\frac{x^m}{\sqrt{ax+b}}}=
\frac{2x^m\sqrt{ax+b}}{(2m+1)a}-\frac{2mb}{(2m+1)a}
~\int{\frac{x^{m-1}}{\sqrt{ax+b}}}
$$
<<*>>=
)clear all

--S 73     14:94 Axiom cannot do this integral
aa:=integrate(x^m/sqrt(a*x+b),x)
--R 
--R
--R           x       m
--I         ++      %L
--I   (1)   |   ----------- d%L
--R        ++    +--------+
--I             \|b + %L a
--R                                          Type: Union(Expression Integer,...)
--E 
@

\section{\cite{1}:14.95~~~~~$\displaystyle
\int{\frac{dx}{x^m\sqrt{ax+b}}}$}
$$\int{\frac{1}{x^m\sqrt{ax+b}}}=
-\frac{\sqrt{ax+b}}{(m-1)bx^{m-1}}-\frac{(2m-3)a}{(2m-2)b}
~\int{\frac{1}{x^{m-1}\sqrt{ax+b}}}
$$
<<*>>=
)clear all

--S 74     14:95 Axiom cannot do this integral
aa:=integrate(1/(x^m*sqrt(a*x+b)),x)
--R 
--R
--R           x
--R         ++         1
--I   (1)   |   -------------- d%L
--R        ++     m +--------+
--I             %L \|b + %L a
--R                                          Type: Union(Expression Integer,...)
--E 
@

\section{\cite{1}:14.96~~~~~$\displaystyle
\int{x^m\sqrt{ax+b}~dx}$}
$$\int{x^m\sqrt{ax+b}}=
\frac{2x^m}{(2m+3)a}(ax+b)^{3/2}
-\frac{2mb}{(2m+3)a}~\int{x^{m-1}\sqrt{ax+b}}
$$
<<*>>=
)clear all

--S 75     14:96 Axiom cannot do this integral
aa:=integrate(x^m*sqrt(a*x+b),x)
--R 
--R
--R           x
--R         ++    m +--------+
--I   (1)   |   %L \|b + %L a d%L
--R        ++
--R                                          Type: Union(Expression Integer,...)
--E 
@

\section{\cite{1}:14.97~~~~~$\displaystyle
\int{\frac{\sqrt{ax+b}}{x^m}~dx}$}
$$\int{\frac{\sqrt{ax+b}}{x^m}}=
-\frac{\sqrt{ax+b}}{(m-1)x^{m-1}}
+\frac{a}{2(m-1)}~\int{\frac{1}{x^{m-1}\sqrt{ax+b}}}
$$
<<*>>=
)clear all

--S 76     14:97 Axiom cannot do this integral
aa:=integrate(sqrt(a*x+b)/x^m,x)
--R 
--R
--R           x  +--------+
--I         ++  \|b + %L a
--I   (1)   |   ----------- d%L
--R        ++         m
--I                 %L
--R                                          Type: Union(Expression Integer,...)
--E 
@

\section{\cite{1}:14.98~~~~~$\displaystyle
\int{\frac{\sqrt{ax+b}}{x^m}~dx}$}
$$\int{\frac{\sqrt{ax+b}}{x^m}}=
\frac{-(ax+b)^{3/2}}{(m-1)bx^{m-1}}
-\frac{(2m-5)a}{(2m-2)b}~\int{\frac{\sqrt{ax+b}}{x^{m-1}}}
$$
Note: 14.98 is the same as 14.97
<<*>>=
)clear all

--S 77     14:98 Axiom cannot do this integral
aa:=integrate(sqrt(a*x+b)/x^m,x)
--R 
--R
--R           x  +--------+
--I         ++  \|b + %L a
--I   (1)   |   ----------- d%L
--R        ++         m
--I                 %L
--R                                          Type: Union(Expression Integer,...)
--E 
@

\section{\cite{1}:14.99~~~~~$\displaystyle
\int{(ax+b)^{m/2}~dx}$}
$$\int{(ax+b)^{m/2}}=
\frac{2(ax+b)^{(m+2)/2}}{a(m+2)}
$$
<<*>>=
)clear all

--S 78
aa:=integrate((a*x+b)^(m/2),x)
--R 
--R
--R                     m log(a x + b)
--R                     --------------
--R                            2
--R        (2a x + 2b)%e
--R   (1)  ---------------------------
--R                  a m + 2a
--R                                          Type: Union(Expression Integer,...)
--E 

--S 79
bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2))
--R 
--R
--R                  m + 2
--R                  -----
--R                    2
--R        2(a x + b)
--R   (2)  ---------------
--R            a m + 2a
--R                                                     Type: Expression Integer
--E

--S 80
cc:=aa-bb
--R 
--R
--R                     m log(a x + b)             m + 2
--R                     --------------             -----
--R                            2                     2
--R        (2a x + 2b)%e               - 2(a x + b)
--R   (3)  ---------------------------------------------
--R                           a m + 2a
--R                                                     Type: Expression Integer
--E

--S 81
explog:=rule(%e^(n*log(x)) == x^n)
--R
--R          n log(x)     n
--R   (4)  %e         == x
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E

--S 82
dd:=explog cc
--R
--R                    m + 2                       m
--R                    -----                       -
--R                      2                         2
--R        - 2(a x + b)      + (2a x + 2b)(a x + b)
--R   (5)  -----------------------------------------
--R                         a m + 2a
--R                                                     Type: Expression Integer
--E

--S 83     14:99 Schaums and Axiom agree
ee:=complexNormalize dd
--R
--R   (6)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.100~~~~~$\displaystyle
\int{x(ax+b)^{m/2}~dx}$}
$$\int{x(ax+b)^{m/2}}=
\frac{2(ax+b)^{(m+4)/2}}{a^2(m+4)}
-\frac{2b(ax+b)^{(m+2)/2}}{a^2(m+2)}
$$
<<*>>=
)clear all

--S 84
aa:=integrate(x*(a*x+b)^(m/2),x)
--R 
--R
--R                                           m log(a x + b)
--R                                           --------------
--R            2      2  2                2          2
--R        ((2a m + 4a )x  + 2a b m x - 4b )%e
--R   (1)  -------------------------------------------------
--R                         2 2     2      2
--R                        a m  + 6a m + 8a
--R                                          Type: Union(Expression Integer,...)
--E 

--S 85
bb:=(2*(a*x+b)^((m+4)/2))/(a^2*(m+4))-(2*b*(a*x+b)^((m+2)/2))/(a^2*(m+2))
--R 
--R
--R                         m + 4                         m + 2
--R                         -----                         -----
--R                           2                             2
--R        (2m + 4)(a x + b)      + (- 2b m - 8b)(a x + b)
--R   (2)  ----------------------------------------------------
--R                           2 2     2      2
--R                          a m  + 6a m + 8a
--R                                                     Type: Expression Integer
--E

--S 86
cc:=aa-bb
--R 
--R
--R   (3)
--R                                          m log(a x + b)
--R                                          --------------
--R           2      2  2                2          2
--R       ((2a m + 4a )x  + 2a b m x - 4b )%e
--R     + 
--R                          m + 4                       m + 2
--R                          -----                       -----
--R                            2                           2
--R       (- 2m - 4)(a x + b)      + (2b m + 8b)(a x + b)
--R  /
--R      2 2     2      2
--R     a m  + 6a m + 8a
--R                                                     Type: Expression Integer
--E

--S 87
explog:=rule(%e^(n*log(x)) == x^n)
--R
--R          n log(x)     n
--R   (4)  %e         == x
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E

--S 88
dd:=explog cc
--R
--R   (5)
--R                          m + 4                       m + 2
--R                          -----                       -----
--R                            2                           2
--R       (- 2m - 4)(a x + b)      + (2b m + 8b)(a x + b)
--R     + 
--R                                                 m
--R                                                 -
--R           2      2  2                2          2
--R       ((2a m + 4a )x  + 2a b m x - 4b )(a x + b)
--R  /
--R      2 2     2      2
--R     a m  + 6a m + 8a
--R                                                     Type: Expression Integer
--E

--S 89     14:100 Schaums and Axiom agree
ee:=complexNormalize dd
--R
--R   (6)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.101~~~~~$\displaystyle
\int{x^2(ax+b)^{m/2}~dx}$}
$$\int{x^2(ax+b)^{m/2}}=
\frac{2(ax+b)^{(m+6)/2}}{a^3(m+6)}
-\frac{4b(ax+b)^{(m+4)/2}}{a^3(m+4)}
+\frac{2b^2(ax+b)^{(m+2)/2}}{a^3(m+2)}
$$
<<*>>=
)clear all

--S 90
aa:=integrate(x^2*(a*x+b)^(m/2),x)
--R 
--R
--R   (1)
--R           3 2      3       3  3      2   2     2     2       2         3
--R       ((2a m  + 12a m + 16a )x  + (2a b m  + 4a b m)x  - 8a b m x + 16b )
--R    *
--R         m log(a x + b)
--R         --------------
--R                2
--R       %e
--R  /
--R      3 3      3 2      3       3
--R     a m  + 12a m  + 44a m + 48a
--R                                          Type: Union(Expression Integer,...)
--E 

--S 91
bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_
      (4*b*(a*x+b)^((m+4)/2))/(a^3*(m+4))+_
        (2*b^2*(a*x+b)^((m+2)/2))/(a^3*(m+2))
--R 
--R
--R   (2)
--R                                m + 6                                   m + 4
--R                                -----                                   -----
--R          2                       2            2                          2
--R       (2m  + 12m + 16)(a x + b)      + (- 4b m  - 32b m - 48b)(a x + b)
--R     + 
--R                                      m + 2
--R                                      -----
--R          2 2      2       2            2
--R       (2b m  + 20b m + 48b )(a x + b)
--R  /
--R      3 3      3 2      3       3
--R     a m  + 12a m  + 44a m + 48a
--R                                                     Type: Expression Integer
--E

--S 92
cc:=aa-bb
--R 
--R
--R   (3)
--R             3 2      3       3  3      2   2     2     2       2         3
--R         ((2a m  + 12a m + 16a )x  + (2a b m  + 4a b m)x  - 8a b m x + 16b )
--R      *
--R           m log(a x + b)
--R           --------------
--R                  2
--R         %e
--R     + 
--R                                  m + 6                                 m + 4
--R                                  -----                                 -----
--R            2                       2          2                          2
--R       (- 2m  - 12m - 16)(a x + b)      + (4b m  + 32b m + 48b)(a x + b)
--R     + 
--R                                        m + 2
--R                                        -----
--R            2 2      2       2            2
--R       (- 2b m  - 20b m - 48b )(a x + b)
--R  /
--R      3 3      3 2      3       3
--R     a m  + 12a m  + 44a m + 48a
--R                                                     Type: Expression Integer
--E

--S 93
explog:=rule(%e^(n*log(x)) == x^n)
--R
--R          n log(x)     n
--R   (4)  %e         == x
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E

--S 94
dd:=explog cc
--R
--R   (5)
--R                                  m + 6                                 m + 4
--R                                  -----                                 -----
--R            2                       2          2                          2
--R       (- 2m  - 12m - 16)(a x + b)      + (4b m  + 32b m + 48b)(a x + b)
--R     + 
--R                                        m + 2
--R                                        -----
--R            2 2      2       2            2
--R       (- 2b m  - 20b m - 48b )(a x + b)
--R     + 
--R             3 2      3       3  3      2   2     2     2       2         3
--R         ((2a m  + 12a m + 16a )x  + (2a b m  + 4a b m)x  - 8a b m x + 16b )
--R      *
--R                  m
--R                  -
--R                  2
--R         (a x + b)
--R  /
--R      3 3      3 2      3       3
--R     a m  + 12a m  + 44a m + 48a
--R                                                     Type: Expression Integer
--E

--S 95     14:101 Schaums and Axiom agree
ee:=complexNormalize dd
--R
--R   (6)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.102~~~~~$\displaystyle
\int{\frac{(ax+b)^{m/2}}{x}~dx}$}
$$\int{\frac{(ax+b)^{m/2}}{x}}=
\frac{2(ax+b)^{m/2}}{m}
+b~\int{\frac{(ax+b)^{(m-2)/2}}{x}}
$$
<<*>>=
)clear all

--S 96     14:102 Axiom cannot do this integral
aa:=integrate((a*x+b)^(m/2)/x,x)
--R 
--R
--R                       m
--R                       -
--R           x           2
--I         ++  (b + %L a)
--I   (1)   |   ----------- d%L
--I        ++        %L
--R                                          Type: Union(Expression Integer,...)
--E 
@
\section{\cite{1}:14.103~~~~~$\displaystyle
\int{\frac{(ax+b)^{m/2}}{x^2}~dx}$}
$$\int{\frac{(ax+b)^{m/2}}{x^2}}=
-\frac{(ax+b)^{(m+2)/2}}{bx}
+\frac{ma}{2b}~\int{\frac{(ax+b)^{m/2}}{x}}
$$
<<*>>=
)clear all

--S 97     14:103 Axiom cannot do this integral
aa:=integrate((a*x+b)^(m/2)/x^2,x)
--R 
--R
--R                       m
--R                       -
--R           x           2
--I         ++  (b + %L a)
--I   (1)   |   ----------- d%L
--R        ++         2
--I                 %L
--R                                          Type: Union(Expression Integer,...)
--E 
@
\section{\cite{1}:14.104~~~~~$\displaystyle
\int{\frac{dx}{x(ax+b)^{m/2}}}$}
$$\int{\frac{1}{x(ax+b)^{m/2}}}=
\frac{2}{(m-2)b(ax+b)^{(m-2)/2}}
+\frac{1}{b}~\int{\frac{1}{x(ax+b)^{(m-2)/2}}}
$$
<<*>>=
)clear all

--S 98     14:104 Axiom cannot do this integral
aa:=integrate(1/(x*(a*x+b)^(m/2)),x)
--R 
--R
--R           x
--R         ++         1
--I   (1)   |   -------------- d%L
--R        ++                m
--R                          -
--R                          2
--I             %L (b + %L a)
--R                                          Type: Union(Expression Integer,...)
--E 

)spool
)lisp (bye)
@

\eject
\begin{thebibliography}{99}
\bibitem{1} Spiegel, Murray R.
{\sl Mathematical Handbook of Formulas and Tables}\\
Schaum's Outline Series McGraw-Hill 1968 pp61-62
\end{thebibliography}
\end{document}