\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/input schaum1.input}
\author{Timothy Daly}
\maketitle
\eject
\tableofcontents
\eject
\section{\cite{1}:14.59~~~~~$\displaystyle
\int{\frac{dx}{ax+b}}$}
$$\int{\frac{1}{ax+b}}=
\frac{1}{a}~\ln(ax+b)
$$
<<*>>=
)spool schaum1.output
)set message test on
)set message auto off
)clear all

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

--S 2
bb:=1/a*log(a*x+b)
--R
--R        log(a x + b)
--R   (2)  ------------
--R              a
--R                                                     Type: Expression Integer
--E

--S 3      14:59 Schaums and Axiom agree
cc:=bb-aa
--R
--R   (3)  0
--R                                                     Type: Expression Integer
--E
@
\section{\cite{1}:14.60~~~~~$\displaystyle
\int{\frac{x~dx}{ax+b}}$}
$$\int{\frac{x}{ax+b}}=
\frac{x}{a}-\frac{b}{a^2}~\ln(ax+b)
$$
<<*>>=
)clear all

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

--S 5
bb:=x/a-b/a^2*log(a*x+b)
--R
--R        - b log(a x + b) + a x
--R   (2)  ----------------------
--R                   2
--R                  a
--R                                                     Type: Expression Integer
--E

--S 6      14:60 Schaums and Axiom agree
cc:=bb-aa
--R
--R   (3)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.61~~~~~$\displaystyle
\int{\frac{x^2~dx}{ax+b}}$}
$$\int{\frac{x^2}{ax+b}}=
\frac{(ax+b)^2}{2a^3}-\frac{2b(ax+b)}{a^3}+\frac{b^2}{a^3}~\ln(ax+b)
$$
<<*>>=
)clear all

--S 7
aa:=integrate(x^2/(a*x+b),x)
--R
--R          2                2 2
--R        2b log(a x + b) + a x  - 2a b x
--R   (1)  -------------------------------
--R                        3
--R                      2a
--R                                          Type: Union(Expression Integer,...)
--E 

--S 8
bb:=(a*x+b)^2/(2*a^3)-(2*b*(a*x+b))/a^3+b^2/a^3*log(a*x+b)
--R
--R          2                2 2              2
--R        2b log(a x + b) + a x  - 2a b x - 3b
--R   (2)  -------------------------------------
--R                           3
--R                         2a
--R                                                     Type: Expression Integer
--E

--S 9
cc:=bb-aa
--R
--R            2
--R          3b
--R   (3)  - ---
--R            3
--R          2a
--R                                                     Type: Expression Integer
--E
@
This factor is constant with respect to $x$ as shown by taking the
derivative. It is a constant of integration.
<<*>>=
--S 10     14:61 Schaums and Axiom differ by a constant
differentiate(cc,x)
--R
--R   (4)  0
--R                                                     Type: Expression Integer
--E
@
\section{\cite{1}:14.62~~~~~$\displaystyle
\int{\frac{x^3~dx}{ax+b}}$}
$$\int{\frac{x^3}{ax+b}}=
\frac{(ax+b)^3}{3a^4}-\frac{3b(ax+b)^2}{2a^4}+
\frac{3b^2(ax+b)}{a^4}-\frac{b^3}{a^4}~\ln(ax+b)
$$
<<*>>=
)clear all

--S 11
aa:=integrate(x^3/(a*x+b),x)
--R
--R            3                 3 3     2   2       2
--R        - 6b log(a x + b) + 2a x  - 3a b x  + 6a b x
--R   (1)  --------------------------------------------
--R                               4
--R                             6a
--R                                          Type: Union(Expression Integer,...)
--E
@
and the book expression is:
<<*>>=
--S 12
bb:=(a*x+b)^3/(3*a^4)-(3*b*(a*x+b)^2)/(2*a^4)+(3*b^2*(a*x+b))/a^4-(b^3/a^4)*log(a*x+b)
--R
--R            3                 3 3     2   2       2       3
--R        - 6b log(a x + b) + 2a x  - 3a b x  + 6a b x + 11b
--R   (2)  ---------------------------------------------------
--R                                  4
--R                                6a
--R                                                     Type: Expression Integer
--E 
@

The difference is a constant with respect to x:
<<*>>=
--S 13
cc:=aa-bb
--R
--R             3
--R          11b
--R   (3)  - ----
--R             4
--R           6a
--R                                                     Type: Expression Integer
--E 
@

If we differentiate each expression we see that this is the integration
constant.
<<*>>=
--S 14     14:62 Schaums and Axiom differ by a constant
dd:=D(cc,x)
--R
--R   (4)  0
--R                                                     Type: Expression Integer
--E 
@

\section{\cite{1}:14.63~~~~~$\displaystyle
\int{\frac{dx}{x~(ax+b)}}$}
$$\int{\frac{1}{x~(ax+b)}}=
\frac{1}{b}~\ln\left(\frac{x}{ax+b}\right)
$$
<<*>>=
)clear all

--S 15
aa:=integrate(1/(x*(a*x+b)),x)
--R
--R        - log(a x + b) + log(x)
--R   (1)  -----------------------
--R                   b
--R                                          Type: Union(Expression Integer,...)
--E 

--S 16
bb:=1/b*log(x/(a*x+b))
--R
--R               x
--R        log(-------)
--R            a x + b
--R   (2)  ------------
--R              b
--R                                                     Type: Expression Integer
--E

--S 17
cc:=aa-bb
--R
--R                                         x
--R        - log(a x + b) + log(x) - log(-------)
--R                                      a x + b
--R   (3)  --------------------------------------
--R                           b
--R                                                     Type: Expression Integer
--E
@
but we know that $$\log(a)-\log(b)=\log(\frac{a}{b})$$

We can express this fact as a rule:
<<*>>=
--S 18
logdiv:=rule(log(a)-log(b) == log(a/b))
--R
--R                                      a
--I   (4)  - log(b) + log(a) + %I == log(-) + %I
--R                                      b
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E 
@
and use this rule to rewrite the logs into divisions:
<<*>>=
--S 19     14:63 Schaums and Axiom agree
dd:=logdiv cc
--R
--R   (5)  0
--R                                                     Type: Expression Integer
--E
@
so we can see the equivalence directly.

\section{\cite{1}:14.64~~~~~$\displaystyle
\int{\frac{dx}{x^2~(ax+b)}}$}
$$\int{\frac{1}{x^2~(ax+b)}}=
-\frac{1}{bx}+\frac{a}{b^2}~\ln\left(\frac{ax+b}{x}\right)
$$
<<*>>=
)clear all

--S 20
aa:=integrate(1/(x^2*(a*x+b)),x)
--R
--R        a x log(a x + b) - a x log(x) - b
--R   (1)  ---------------------------------
--R                        2
--R                       b x
--R                                          Type: Union(Expression Integer,...)
--E 
@

The original form given in the book expands to:
<<*>>=
--S 21
bb:=-1/(b*x)+a/b^2*log((a*x+b)/x)
--R
--R                a x + b
--R        a x log(-------) - b
--R                   x
--R   (2)  --------------------
--R                  2
--R                 b x
--R                                                     Type: Expression Integer
--E 

--S 22
cc:=aa-bb
--R
--R                                          a x + b
--R        a log(a x + b) - a log(x) - a log(-------)
--R                                             x
--R   (3)  ------------------------------------------
--R                             2
--R                            b
--R                                                     Type: Expression Integer
--E
@

We can define the following rule to expand log forms:
<<*>>=
--S 23
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R            a
--R   (4)  log(-) == - log(b) + log(a)
--R            b
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E 
@
and apply it to the difference
<<*>>=
--S 24     14:64 Schaums and Axiom agree
divlog cc
--R
--R   (5)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.65~~~~~$\displaystyle
\int{\frac{dx}{x^3~(ax+b)}}$}
$$\int{\frac{1}{x^3~(ax+b)}}=
\frac{2ax-b}{2b^2x^2}+\frac{a^2}{b^3}~\ln\left(\frac{x}{ax+b}\right)
$$
<<*>>=
)clear all
--S 25
aa:=integrate(1/(x^3*(a*x+b)),x)
--R
--R            2 2                 2 2                   2
--R        - 2a x log(a x + b) + 2a x log(x) + 2a b x - b
--R   (1)  -----------------------------------------------
--R                               3 2
--R                             2b x
--R                                          Type: Union(Expression Integer,...)
--E

--S 26
bb:=(2*a*x-b)/(2*b^2*x^2)+a^2/b^3*log(x/(a*x+b))
--R
--R          2 2       x                 2
--R        2a x log(-------) + 2a b x - b
--R                 a x + b
--R   (2)  -------------------------------
--R                       3 2
--R                     2b x
--R                                                     Type: Expression Integer
--E

--S 27
cc:=aa-bb
--R
--R           2                2          2       x
--R        - a log(a x + b) + a log(x) - a log(-------)
--R                                            a x + b
--R   (3)  --------------------------------------------
--R                              3
--R                             b
--R                                                     Type: Expression Integer
--E

--S 28
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R            a
--R   (4)  log(-) == - log(b) + log(a)
--R            b
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E

--S 29     14:65 Schaums and Axiom agree
dd:=divlog cc
--R
--R   (5)  0
--R                                                     Type: Expression Integer
--E 
@

\section{\cite{1}:14.66~~~~~$\displaystyle
\int{\frac{dx}{(ax+b)^2}}$}
$$\int{\frac{1}{(ax+b)^2}}=
\frac{-1}{a~(ax+b)}
$$
<<*>>=
)clear all

--S 30
aa:=integrate(1/(a*x+b)^2,x)
--R
--R              1
--R   (1)  - ---------
--R           2
--R          a x + a b
--R                                          Type: Union(Expression Integer,...)
--E 

--S 31
bb:=-1/(a*(a*x+b))
--R
--R              1
--R   (2)  - ---------
--R           2
--R          a x + a b
--R                                            Type: Fraction Polynomial Integer
--E

--S 32     14:66 Schaums and Axiom agree
cc:=aa-bb
--R
--R   (3)  0
--R                                                     Type: Expression Integer
--E

@

\section{\cite{1}:14.67~~~~~$\displaystyle
\int{\frac{x~dx}{(ax+b)^2}}$}
$$\int{\frac{x}{(ax+b)^2}}=
\frac{b}{a^2~(ax+b)}+\frac{1}{a^2}~\ln(ax+b)
$$
<<*>>=
)clear all

--S 33
aa:=integrate(x/(a*x+b)^2,x)
--R
--R        (a x + b)log(a x + b) + b
--R   (1)  -------------------------
--R                 3     2
--R                a x + a b
--R                                          Type: Union(Expression Integer,...)
--E 

--S 34
bb:=b/(a^2*(a*x+b))+1/a^2*log(a*x+b)
--R
--R        (a x + b)log(a x + b) + b
--R   (2)  -------------------------
--R                 3     2
--R                a x + a b
--R                                                     Type: Expression Integer
--E

--S 35     14:67 Schaums and Axiom agree
cc:=aa-bb
--R
--R   (3)  0
--R                                                     Type: Expression Integer
--E

@

\section{\cite{1}:14.68~~~~~$\displaystyle
\int{\frac{x^2~dx}{(ax+b)^2}}$}
$$\int{\frac{x^2}{(ax+b)^2}}=
\frac{ax+b}{a^3}-\frac{b^2}{a^3~(ax+b)}
-\frac{2b}{a^3}~\ln(ax+b)
$$
<<*>>=
)clear all

--S 36
aa:=integrate(x^2/(a*x+b)^2,x)
--R
--R                      2                 2 2            2
--R        (- 2a b x - 2b )log(a x + b) + a x  + a b x - b
--R   (1)  ------------------------------------------------
--R                             4     3
--R                            a x + a b
--R                                          Type: Union(Expression Integer,...)
--E 
@
and the book expression expands into
<<*>>=
--S 37
bb:=(a*x+b)/a^3-b^2/(a^3*(a*x+b))-((2*b)/a^3)*log(a*x+b)
--R
--R                      2                 2 2
--R        (- 2a b x - 2b )log(a x + b) + a x  + 2a b x
--R   (2)  --------------------------------------------
--R                           4     3
--R                          a x + a b
--R                                                     Type: Expression Integer
--E 
@

These two expressions differ by the constant
<<*>>=
--S 38
cc:=aa-bb
--R
--R           b
--R   (3)  - --
--R           3
--R          a
--R                                                     Type: Expression Integer
--E 
@

That this expression is constant can be shown by differentiation:
<<*>>=
--S 39     14:68 Schaums and Axiom differ by a constant
D(cc,x)
--R
--R   (4)  0
--R                                                     Type: Expression Integer
--E 
@

\section{\cite{1}:14.69~~~~~$\displaystyle
\int{\frac{x^3~dx}{(ax+b)^2}}$}
$$\int{\frac{x^3}{(ax+b)^2}}=
\frac{(ax+b)^2}{2a^4}-\frac{3b(ax+b)}{a^4}+\frac{b^3}{a^4(ax+b)}
+\frac{3b^2}{a^4}~\ln(ax+b)
$$
<<*>>=
)clear all

--S 40
aa:=integrate(x^3/(a*x+b)^2,x)
--R
--R             2      3                 3 3     2   2       2      3
--R        (6a b x + 6b )log(a x + b) + a x  - 3a b x  - 4a b x + 2b
--R   (1)  ----------------------------------------------------------
--R                                  5      4
--R                                2a x + 2a b
--R                                          Type: Union(Expression Integer,...)
--E

--S 41
bb:=(a*x+b)^2/(2*a^4)-(3*b*(a*x+b))/a^4+b^3/(a^4*(a*x+b))+(3*b^2/a^4)*log(a*x+b)
--R
--R             2      3                 3 3     2   2       2      3
--R        (6a b x + 6b )log(a x + b) + a x  - 3a b x  - 9a b x - 3b
--R   (2)  ----------------------------------------------------------
--R                                  5      4
--R                                2a x + 2a b
--R                                                     Type: Expression Integer
--E

--S 42
cc:=aa-bb
--R
--R          2
--R        5b
--R   (3)  ---
--R          4
--R        2a
--R                                                     Type: Expression Integer
--E

--S 43     14:69 Schaums and Axiom differ by a constant
dd:=D(cc,x)
--R
--R   (4)  0
--R                                                     Type: Expression Integer
--E
@
\section{\cite{1}:14.70~~~~~$\displaystyle
\int{\frac{dx}{x~(ax+b)^2}}$}
$$\int{\frac{1}{x~(ax+b)^2}}=
\frac{1}{b~(ax+b)}+\frac{1}{b^2}~\ln\left(\frac{x}{ax+b}\right)
$$
<<*>>=
)clear all

--S 44
aa:=integrate(1/(x*(a*x+b)^2),x)
--R
--R        (- a x - b)log(a x + b) + (a x + b)log(x) + b
--R   (1)  ---------------------------------------------
--R                             2     3
--R                          a b x + b
--R                                          Type: Union(Expression Integer,...)
--E
@
and the book says:
<<*>>=
--S 45
bb:=(1/(b*(a*x+b))+(1/b^2)*log(x/(a*x+b)))
--R
--R                        x
--R        (a x + b)log(-------) + b
--R                     a x + b
--R   (2)  -------------------------
--R                   2     3
--R                a b x + b
--R                                                     Type: Expression Integer
--E

--S 46
cc:=aa-bb
--R
--R                                         x
--R        - log(a x + b) + log(x) - log(-------)
--R                                      a x + b
--R   (3)  --------------------------------------
--R                           2
--R                          b
--R                                                     Type: Expression Integer
--E
@
So we look at the divlog rule again:
<<*>>=
--S 47
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R            a
--R   (4)  log(-) == - log(b) + log(a)
--R            b
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E
@

we apply it:
<<*>>=
--S 48     14:70 Schaums and Axiom agree
dd:=divlog cc
--R
--R   (5)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.71~~~~~$\displaystyle
\int{\frac{dx}{x^2~(ax+b)^2}}$}
$$\int{\frac{1}{x^2~(ax+b)^2}}=
\frac{-a}{b^2~(ax+b)}-\frac{1}{b^2~x}+
\frac{2a}{b^3}~\ln\left(\frac{ax+b}{x}\right)
$$
<<*>>=
)clear all

--S 49
aa:=integrate(1/(x^2*(a*x+b)^2),x)
--R
--R           2 2                              2 2                             2
--R        (2a x  + 2a b x)log(a x + b) + (- 2a x  - 2a b x)log(x) - 2a b x - b
--R   (1)  ---------------------------------------------------------------------
--R                                        3 2    4
--R                                     a b x  + b x
--R                                          Type: Union(Expression Integer,...)
--E
@
and the book says:
<<*>>=
--S 50
bb:=(-a/(b^2*(a*x+b)))-(1/(b^2*x))+((2*a)/b^3)*log((a*x+b)/x)
--R
--R           2 2              a x + b              2
--R        (2a x  + 2a b x)log(-------) - 2a b x - b
--R                               x
--R   (2)  ------------------------------------------
--R                          3 2    4
--R                       a b x  + b x
--R                                                     Type: Expression Integer
--E

--S 51
cc:=aa-bb
--R
--R                                             a x + b
--R        2a log(a x + b) - 2a log(x) - 2a log(-------)
--R                                                x
--R   (3)  ---------------------------------------------
--R                               3
--R                              b
--R                                                     Type: Expression Integer
--E
@
which calls for our divlog rule:
<<*>>=
--S 52
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R            a
--R   (4)  log(-) == - log(b) + log(a)
--R            b
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E
@
which we use to transform the result:
<<*>>=
--S 53     14:71 Schaums and Axiom agree
dd:=divlog cc
--R
--R   (5)  0
--R                                                     Type: Expression Integer
--E
@
\section{\cite{1}:14.72~~~~~$\displaystyle
\int{\frac{dx}{x^3~(ax+b)^2}}$}
$$\int{\frac{1}{x^3~(ax+b)^2}}=
-\frac{(ax+b)^2}{2b^4x^2}+\frac{3a(ax+b)}{b^4x}-
\frac{a^3x}{b^4(ax+b)}-\frac{3a^2}{b^4}~\ln\left(\frac{ax+b}{x}\right)
$$
<<*>>=
)clear all

--S 54
aa:=integrate(1/(x^3*(a*x+b)^2),x)
--R
--R   (1)
--R            3 3     2   2                   3 3     2   2            2   2
--R       (- 6a x  - 6a b x )log(a x + b) + (6a x  + 6a b x )log(x) + 6a b x
--R     + 
--R           2     3
--R       3a b x - b
--R  /
--R         4 3     5 2
--R     2a b x  + 2b x
--R                                          Type: Union(Expression Integer,...)
--E

--S 55
bb:=-(a*x+b)^2/(2*b^4*x^2)+(3*a*(a*x+b))/(b^4*x)-(a^3*x)/(b^4*(a*x+b))-((3*a^2)/b^4)*log((a*x+b)/x)
--R
--R             3 3     2   2     a x + b      3 3     2   2       2     3
--R        (- 6a x  - 6a b x )log(-------) + 3a x  + 9a b x  + 3a b x - b
--R                                  x
--R   (2)  ---------------------------------------------------------------
--R                                    4 3     5 2
--R                                2a b x  + 2b x
--R                                                     Type: Expression Integer
--E

--S 56
cc:=aa-bb
--R
--R            2                 2           2    a x + b      2
--R        - 6a log(a x + b) + 6a log(x) + 6a log(-------) - 3a
--R                                                  x
--R   (3)  -----------------------------------------------------
--R                                   4
--R                                 2b
--R                                                     Type: Expression Integer
--E

--S 57
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R            a
--R   (4)  log(-) == - log(b) + log(a)
--R            b
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E

--S 58
dd:=divlog cc
--R
--R            2
--R          3a
--R   (5)  - ---
--R            4
--R          2b
--R                                                     Type: Expression Integer
--E

--S 59     14:72 Schaums and Axiom differ by a constant
ee:=D(dd,x)
--R
--R   (6)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.73~~~~~$\displaystyle
\int{\frac{dx}{(ax+b)^3}}$}
$$\int{\frac{1}{(ax+b)^3}}=
\frac{-1}{2a(ax+b)^2}
$$
<<*>>=
)clear all

--S 60
aa:=integrate(1/(a*x+b)^3,x)
--R
--R                     1
--R   (1)  - ----------------------
--R            3 2     2          2
--R          2a x  + 4a b x + 2a b
--R                                          Type: Union(Expression Integer,...)
--E

--S 61
bb:=-1/(2*(a*x+b)^2)
--R
--R                    1
--R   (2)  - --------------------
--R            2 2              2
--R          2a x  + 4a b x + 2b
--R                                            Type: Fraction Polynomial Integer
--E

--S 62
cc:=aa-bb
--R
--R                 a - 1
--R   (3)  ----------------------
--R          3 2     2          2
--R        2a x  + 4a b x + 2a b
--R                                                     Type: Expression Integer
--E

--S 63
dd:=aa/bb
--R
--R        1
--R   (4)  -
--R        a
--R                                                     Type: Expression Integer
--E

--S 64     14:73 Schaums and Axiom differ by a constant
ee:=D(dd,x)
--R
--R   (5)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.74~~~~~$\displaystyle
\int{\frac{x~dx}{(ax+b)^3}}$}
$$\int{\frac{x}{(ax+b)^3}}=
\frac{-1}{a^2(ax+b)}+\frac{b}{2a^2(ax+b)^2}
$$
<<*>>=
)clear all

--S 65
aa:=integrate(x/(a*x+b)^3,x)
--R
--R              - 2a x - b
--R   (1)  ----------------------
--R          4 2     3        2 2
--R        2a x  + 4a b x + 2a b
--R                                          Type: Union(Expression Integer,...)
--E

--S 66
bb:=-1/(a^2*(a*x+b))+b/(2*a^2*(a*x+b)^2)
--R
--R              - 2a x - b
--R   (2)  ----------------------
--R          4 2     3        2 2
--R        2a x  + 4a b x + 2a b
--R                                            Type: Fraction Polynomial Integer
--E

--S 67     14:74 Schaums and Axiom agree
cc:=aa-bb
--R
--R   (3)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.75~~~~~$\displaystyle
\int{\frac{x^2~dx}{(ax+b)^3}}$}
$$\int{\frac{x^2}{(ax+b)^3}}=
\frac{2b}{a^3(ax+b)}-\frac{b^2}{2a^3(ax+b)^2}+
\frac{1}{a^3}~\ln(ax+b)
$$
<<*>>=
)clear all

--S 68
aa:=integrate(x^2/(a*x+b)^3,x)
--R
--R           2 2              2                           2
--R        (2a x  + 4a b x + 2b )log(a x + b) + 4a b x + 3b
--R   (1)  -------------------------------------------------
--R                        5 2     4        3 2
--R                      2a x  + 4a b x + 2a b
--R                                          Type: Union(Expression Integer,...)
--E

--S 69
bb:=(2*b)/(a^3*(a*x+b))-(b^2)/(2*a^3*(a*x+b)^2)+1/a^3*log(a*x+b)
--R
--R           2 2              2                           2
--R        (2a x  + 4a b x + 2b )log(a x + b) + 4a b x + 3b
--R   (2)  -------------------------------------------------
--R                        5 2     4        3 2
--R                      2a x  + 4a b x + 2a b
--R                                                     Type: Expression Integer
--E

--S 70     14:75 Schaums and Axiom agree
cc:=aa-bb
--R
--R   (3)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.76~~~~~$\displaystyle
\int{\frac{x^3~dx}{(ax+b)^3}}$}
$$\int{\frac{x^3}{(ax+b)^3}}=
\frac{x}{a^3}-\frac{3b^2}{a^4(ax+b)}+\frac{b^3}{2a^4(ax+b)^2}-
\frac{3b}{a^4}~\ln(ax+b)
$$
<<*>>=
)clear all
--S 71
aa:=integrate(x^3/(a*x+b)^3,x)
--R
--R   (1)
--R        2   2        2      3                  3 3     2   2       2      3
--R   (- 6a b x  - 12a b x - 6b )log(a x + b) + 2a x  + 4a b x  - 4a b x - 5b
--R   ------------------------------------------------------------------------
--R                              6 2     5        4 2
--R                            2a x  + 4a b x + 2a b
--R                                          Type: Union(Expression Integer,...)
--E

--S 72
bb:=(x/a^3)-(3*b^2)/(a^4*(a*x+b))+b^3/(2*a^4*(a*x+b)^2)-(3*b)/a^4*log(a*x+b)
--R
--R   (2)
--R        2   2        2      3                  3 3     2   2       2      3
--R   (- 6a b x  - 12a b x - 6b )log(a x + b) + 2a x  + 4a b x  - 4a b x - 5b
--R   ------------------------------------------------------------------------
--R                              6 2     5        4 2
--R                            2a x  + 4a b x + 2a b
--R                                                     Type: Expression Integer
--E

--S 73     14:76 Schaums and Axiom agree
cc:=aa-bb
--R
--R   (3)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.77~~~~~$\displaystyle
\int{\frac{dx}{x(ax+b)^3}}$}
$$\int{\frac{1}{x(ax+b)^3}}=
\frac{3}{2b(ax+b)^2}+\frac{2ax}{2b^2(ax+b)^2}-
\frac{1}{b^3}*\ln\left(\frac{ax+b}{x}\right)
$$

<<*>>=
)clear all

--S 74
aa:=integrate(1/(x*(a*x+b)^3),x)
--R
--R   (1)
--R            2 2              2                   2 2              2
--R       (- 2a x  - 4a b x - 2b )log(a x + b) + (2a x  + 4a b x + 2b )log(x)
--R     + 
--R                  2
--R       2a b x + 3b
--R  /
--R       2 3 2       4      5
--R     2a b x  + 4a b x + 2b
--R                                          Type: Union(Expression Integer,...)
--E

--S 75
bb:=(a^2*x^2)/(2*b^3*(a*x+b)^2)-(2*a*x)/(b^3*(a*x+b))-(1/b^3)*log((a*x+b)/x)
--R
--R             2 2              2     a x + b      2 2
--R        (- 2a x  - 4a b x - 2b )log(-------) - 3a x  - 4a b x
--R                                       x
--R   (2)  -----------------------------------------------------
--R                          2 3 2       4      5
--R                        2a b x  + 4a b x + 2b
--R                                                     Type: Expression Integer
--E

--S 76
cc:=aa-bb
--R
--R                                         a x + b
--R        - 2log(a x + b) + 2log(x) + 2log(-------) + 3
--R                                            x
--R   (3)  ---------------------------------------------
--R                               3
--R                             2b
--R                                                     Type: Expression Integer
--E

--S 77
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R            a
--R   (4)  log(-) == - log(b) + log(a)
--R            b
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E

--S 78
dd:=divlog cc
--R
--R         3
--R   (5)  ---
--R          3
--R        2b
--R                                                     Type: Expression Integer
--E

--S 79     14:77 Schaums and Axiom differ by a constant
ee:=D(dd,x)
--R
--R   (6)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.78~~~~~$\displaystyle
\int{\frac{dx}{x^2(ax+b)^3}}$}
$$\int{\frac{1}{x^2(ax+b)^3}}=
\frac{-a}{2b^2(ax+b)^2}-\frac{2a}{b^3(ax+b)}-
\frac{1}{b^3x}+\frac{3a}{b^4}~\ln\left(\frac{ax+b}{x}\right)
$$
<<*>>=
)clear all

--S 80
aa:=integrate(1/(x^2*(a*x+b)^3),x)
--R
--R   (1)
--R          3 3      2   2       2
--R       (6a x  + 12a b x  + 6a b x)log(a x + b)
--R     + 
--R            3 3      2   2       2             2   2       2      3
--R       (- 6a x  - 12a b x  - 6a b x)log(x) - 6a b x  - 9a b x - 2b
--R  /
--R       2 4 3       5 2     6
--R     2a b x  + 4a b x  + 2b x
--R                                          Type: Union(Expression Integer,...)
--E

--S 81
bb:=-a/(2*b^2*(a*x+b)^2)-(2*a)/(b^3*(a*x+b))-1/(b^3*x)+((3*a)/b^4)*log((a*x+b)/x)
--R
--R           3 3      2   2       2      a x + b      2   2       2      3
--R        (6a x  + 12a b x  + 6a b x)log(-------) - 6a b x  - 9a b x - 2b
--R                                          x
--R   (2)  ----------------------------------------------------------------
--R                              2 4 3       5 2     6
--R                            2a b x  + 4a b x  + 2b x
--R                                                     Type: Expression Integer
--E

--S 82
cc:=aa-bb
--R
--R                                             a x + b
--R        3a log(a x + b) - 3a log(x) - 3a log(-------)
--R                                                x
--R   (3)  ---------------------------------------------
--R                               4
--R                              b
--R                                                     Type: Expression Integer
--E

--S 83
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R            a
--R   (4)  log(-) == - log(b) + log(a)
--R            b
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E

--S 84     14:78 Schaums and Axiom agree
dd:=divlog cc
--R
--R   (5)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.79~~~~~$\displaystyle
\int{\frac{dx}{x^3(ax+b)^3}}$}
$$\int{\frac{1}{x^3(ax+b)^3}}=
-\frac{1}{2bx^2(ax+b)^2}+
\frac{2a}{b^2x(ax+b)^2}+
\frac{9a^2}{b^3(ax+b)^2}+
\frac{6a^3x}{b^4(ax+b)^2}-
\frac{6a^2}{b^5}~\ln\left(\frac{ax+b}{x}\right)$$

<<*>>=
)clear all

--S 85
aa:=integrate(1/(x^3*(a*x+b)^3),x)
--R
--R   (1)
--R             4 4      3   3      2 2 2
--R       (- 12a x  - 24a b x  - 12a b x )log(a x + b)
--R     + 
--R           4 4      3   3      2 2 2             3   3      2 2 2       3     4
--R       (12a x  + 24a b x  + 12a b x )log(x) + 12a b x  + 18a b x  + 4a b x - b
--R  /
--R       2 5 4       6 3     7 2
--R     2a b x  + 4a b x  + 2b x
--R                                          Type: Union(Expression Integer,...)
--E

--S 86
bb:=-1/(2*b*x^2*(a*x+b)^2)_
    +(2*a)/(b^2*x*(a*x+b)^2)_
    +(9*a^2)/(b^3*(a*x+b)^2)_
    +(6*a^3*x)/(b^4*(a*x+b)^2)_
    +(-6*a^2)/b^5*log((a*x+b)/x)
--R
--R   (2)
--R             4 4      3   3      2 2 2     a x + b       3   3      2 2 2
--R       (- 12a x  - 24a b x  - 12a b x )log(-------) + 12a b x  + 18a b x
--R                                              x
--R     + 
--R           3     4
--R       4a b x - b
--R  /
--R       2 5 4       6 3     7 2
--R     2a b x  + 4a b x  + 2b x
--R                                                     Type: Expression Integer
--E

--S 87
cc:=aa-bb
--R
--R            2                 2           2    a x + b
--R        - 6a log(a x + b) + 6a log(x) + 6a log(-------)
--R                                                  x
--R   (3)  -----------------------------------------------
--R                                5
--R                               b
--R                                                     Type: Expression Integer
--E

--S 88
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R            a
--R   (4)  log(-) == - log(b) + log(a)
--R            b
--R                        Type: RewriteRule(Integer,Integer,Expression Integer)
--E

--S 89     14:79 Schaums and Axiom agree
dd:=divlog cc
--R
--R   (5)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.80~~~~~$\displaystyle
\int{(ax+b)^n~dx}$}
$$\int{(ax+b)^n}=
\frac{(ax+b)^{n+1}}{(n+1)a}{\rm\ provided\ }n \ne -1
$$
<<*>>=
)clear all
--S 90
aa:=integrate((a*x+b)^n,x)
--R
--R                   n log(a x + b)
--R        (a x + b)%e
--R   (1)  -------------------------
--R                 a n + a
--R                                          Type: Union(Expression Integer,...)
--E

--S 91
bb:=(a*x+b)^(n+1)/((n+1)*a)
--R
--R                 n + 1
--R        (a x + b)
--R   (2)  --------------
--R            a n + a
--R                                                     Type: Expression Integer
--E

--S 92
cc:=aa-bb
--R
--R                   n log(a x + b)            n + 1
--R        (a x + b)%e               - (a x + b)
--R   (3)  ------------------------------------------
--R                          a n + a
--R                                                     Type: Expression Integer
--E
@
This messy formula can be simplified using the explog rule:
<<*>>=
--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                   n + 1                     n
--R        - (a x + b)      + (a x + b)(a x + b)
--R   (5)  --------------------------------------
--R                        a n + a
--R                                                     Type: Expression Integer
--E

--S 95     14:80 Schaums and Axiom agree
ee:=complexNormalize dd
--R
--R   (6)  0
--R                                                     Type: Expression Integer
--E
@

\section{\cite{1}:14.81~~~~~$\displaystyle
\int{x(ax+b)^n~dx}$}
$$\int{x(ax+b)^n}=
\frac{(ax+b)^{n+2}}{(n+2)a^2}-\frac{b(ax+b)^{n+1}}{(n+1)a^2}
{\rm\ provided\ }n \ne -1,-2
$$
<<*>>=
)clear all
--S 96
aa:=integrate(x*(a*x+b)^n,x)
--R
--R           2     2  2              2   n log(a x + b)
--R        ((a n + a )x  + a b n x - b )%e
--R   (1)  ---------------------------------------------
--R                       2 2     2      2
--R                      a n  + 3a n + 2a
--R                                          Type: Union(Expression Integer,...)
--E

--S 97
bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2)
--R
--R                        n + 2                        n + 1
--R        (n + 1)(a x + b)      + (- b n - 2b)(a x + b)
--R   (2)  --------------------------------------------------
--R                          2 2     2      2
--R                         a n  + 3a n + 2a
--R                                                     Type: Expression Integer
--E

--S 98
cc:=aa-bb
--R
--R   (3)
--R          2     2  2              2   n log(a x + b)                     n + 2
--R       ((a n + a )x  + a b n x - b )%e               + (- n - 1)(a x + b)
--R     + 
--R                          n + 1
--R       (b n + 2b)(a x + b)
--R  /
--R      2 2     2      2
--R     a n  + 3a n + 2a
--R                                                     Type: Expression Integer
--E

--S 99
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 100
dd:=explog cc
--R
--R   (5)
--R                         n + 2                      n + 1
--R       (- n - 1)(a x + b)      + (b n + 2b)(a x + b)
--R     + 
--R          2     2  2              2          n
--R       ((a n + a )x  + a b n x - b )(a x + b)
--R  /
--R      2 2     2      2
--R     a n  + 3a n + 2a
--R                                                     Type: Expression Integer
--E

--S 101
ee:=complexNormalize dd
--R
--R   (6)  0
--R                                                     Type: Expression Integer
--E
@
\section{\cite{1}:14.82~~~~~$\displaystyle
\int{x^2(ax+b)^n~dx}$}
$$\int{x^2(ax+b)^n}=
\frac{(ax+b)^{n+2}}{(n+3)a^3}-
\frac{2b(ax+b)^{n+2}}{(n+2)a^3}+
\frac{b^2(ax+b)^{n+1}}{(n+1)a^3}
{\rm\ provided\ }n \ne -1,-2,-3
$$

<<*>>=
)clear all
--S 102
aa:=integrate(x^2*(a*x+b)^n,x)
--R
--R   (1)
--R      3 2     3      3  3     2   2    2     2       2        3   n log(a x + b)
--R   ((a n  + 3a n + 2a )x  + (a b n  + a b n)x  - 2a b n x + 2b )%e
--R   -----------------------------------------------------------------------------
--R                              3 3     3 2      3      3
--R                             a n  + 6a n  + 11a n + 6a
--R                                          Type: Union(Expression Integer,...)
--E

--S 103
bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1))/((n+1)*a^3)
--R
--R   (2)
--R         2                   n + 3          2                      n + 2
--R       (n  + 3n + 2)(a x + b)      + (- 2b n  - 8b n - 6b)(a x + b)
--R     + 
--R         2 2     2      2          n + 1
--R       (b n  + 5b n + 6b )(a x + b)
--R  /
--R      3 3     3 2      3      3
--R     a n  + 6a n  + 11a n + 6a
--R                                                     Type: Expression Integer
--E

--S 104
cc:=aa-bb
--R
--R   (3)
--R            3 2     3      3  3     2   2    2     2       2        3
--R         ((a n  + 3a n + 2a )x  + (a b n  + a b n)x  - 2a b n x + 2b )
--R      *
--R           n log(a x + b)
--R         %e
--R     + 
--R           2                   n + 3        2                      n + 2
--R       (- n  - 3n - 2)(a x + b)      + (2b n  + 8b n + 6b)(a x + b)
--R     + 
--R           2 2     2      2          n + 1
--R       (- b n  - 5b n - 6b )(a x + b)
--R  /
--R      3 3     3 2      3      3
--R     a n  + 6a n  + 11a n + 6a
--R                                                     Type: Expression Integer
--E

--S 105
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 106
dd:=explog cc
--R
--R   (5)
--R           2                   n + 3        2                      n + 2
--R       (- n  - 3n - 2)(a x + b)      + (2b n  + 8b n + 6b)(a x + b)
--R     + 
--R           2 2     2      2          n + 1
--R       (- b n  - 5b n - 6b )(a x + b)
--R     + 
--R          3 2     3      3  3     2   2    2     2       2        3          n
--R       ((a n  + 3a n + 2a )x  + (a b n  + a b n)x  - 2a b n x + 2b )(a x + b)
--R  /
--R      3 3     3 2      3      3
--R     a n  + 6a n  + 11a n + 6a
--R                                                     Type: Expression Integer
--E

--S 107    14:82 Schaums and Axiom agree
ee:=complexNormalize dd
--R
--R   (6)  0
--R                                                     Type: Expression Integer
--E
@
\section{\cite{1}:14.83~~~~~$\displaystyle
\int{x^m(ax+b)^n}~dx$}
$$\int{x^m(ax+b)^n}
\left\{
\begin{array}{l}
\displaystyle
\frac{x^{m+1}(ax+b)^n}{m+n+1}
+\frac{nb}{m+n+1}\int{x^m(ax+b)^{n-1}}\\
\\
\displaystyle
\frac{x^{m+1}(ax+b)^{n+1}}{(m+n+1)a}
-\frac{mb}{(m+n+1)a}\int{x^{m-1}(ax+b)^n}\\
\\
\displaystyle
\frac{-x^{m+1}(ax+b)^{n+1}}{(n+1)b}
+\frac{m+n+2}{(n+1)b}\int{x^m(ax+b)^{n+1}}\\
\end{array}
\right.
$$

<<*>>=
--S 108    14:83 Axiom cannot do this integration
aa:=integrate(x^m*(a*x+b)^n,x)
--R
--R           x
--R         ++    m          n
--I   (1)   |   %U (b + %U a) d%U
--R        ++
--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 pp60-61
\end{thebibliography}
\end{document}