# [Maxima] Can Maxima do this integral with some additional package?

Karl-Dieter Crisman kcrisman at gmail.com
Fri Mar 9 13:32:05 CST 2012

```This came up in
an optional package which might do more integrals that are HG
functions?

(%i2) display2d:false;

(%o2) false
(%i3) integrate( (a/x + b*x^c)^d, x);

(%o3) 'integrate((b*x^c+a/x)^d,x)

Apparently Mathematica has no trouble with this in terms of
hypergeometric functions.  Here is Maxima, for various values of d.
It can do positive integers, though sometimes with a ghastly series of
questions :)  Thanks!

(%i8)  integrate( (a/x + b*x^c)^(5/4),x);

(%o8) 'integrate((b*x^c+a/x)^(5/4),x)
(%i9)  integrate( (a/x + b*x^c)^(20/3),x);

(%o9) 'integrate((b*x^c+a/x)^(20/3),x)

(%i5) integrate( (a/x + b*x^c)^3,x);

Is  c-1  zero or nonzero?

nonzero;
Is  3*c+1  zero or nonzero?

nonzero;
Is  c  zero or nonzero?

nonzero;
(%o5) b^3*x^(3*c+1)/(3*c+1)+3*a*b^2*x^(2*c)/(2*c)+3*a^2*b*x^(c-1)/(c-1)-a^3
/(2
*x^2)
(%i6) integrate( (a/x + b*x^c)^4,x);

Is  c-2  zero or nonzero?

nonzero;
Is  4*c+1  zero or nonzero?

nonzero;
Is  2*c-1  zero or nonzero?

nonzero;
Is  c  zero or nonzero?

nonzero;
(%o6) b^4*x^(4*c+1)/(4*c+1)+6*a^2*b^2*x^(2*c-1)/(2*c-1)+4*a*b^3*x^(3*c)/(3*c)
+4*a^3*b*x^(c-2)/(c-2)-a^4/(3*x^3)
(%i7) integrate( (a/x + b*x^c)^24,x);

Is  c-22  zero or nonzero?

nonzero;
Is  24*c+1  zero or nonzero?

nonzero;
Is  2*c-21  zero or nonzero?

nonzero;
Is  3*c-20  zero or nonzero?

nonzero;
Is  4*c-19  zero or nonzero?

nonzero;
Is  5*c-18  zero or nonzero?

nonzero;
Is  6*c-17  zero or nonzero?

nonzero;
Is  7*c-16  zero or nonzero?

nonzero;
Is  8*c-15  zero or nonzero?

nonzero;
Is  9*c-14  zero or nonzero?

nonzero;
Is  10*c-13  zero or nonzero?

nonzero;
Is  11*c-12  zero or nonzero?

nonzero;
Is  12*c-11  zero or nonzero?

nonzero;
Is  13*c-10  zero or nonzero?

nonzero;
Is  14*c-9  zero or nonzero?

nonzero;
Is  15*c-8  zero or nonzero?

nonzero;
Is  16*c-7  zero or nonzero?

nonzero;
Is  17*c-6  zero or nonzero?

nonzero;
Is  18*c-5  zero or nonzero?

nonzero;
Is  19*c-4  zero or nonzero?

nonzero;
Is  20*c-3  zero or nonzero?

nonzero;
Is  21*c-2  zero or nonzero?

nonzero;
Is  22*c-1  zero or nonzero?

nonzero;
Is  c  zero or nonzero?

nonzero;
(%o7) b^24*x^(24*c+1)/(24*c+1)+276*a^2*b^22*x^(22*c-1)/(22*c-1)
+2024*a^3*b^21*x^(21*c-2)/(21*c-2)
+10626*a^4*b^20*x^(20*c-3)/(20*c-3)
+42504*a^5*b^19*x^(19*c-4)/(19*c-4)
+134596*a^6*b^18*x^(18*c-5)/(18*c-5)
+346104*a^7*b^17*x^(17*c-6)/(17*c-6)
+735471*a^8*b^16*x^(16*c-7)/(16*c-7)
+1307504*a^9*b^15*x^(15*c-8)/(15*c-8)
+1961256*a^10*b^14*x^(14*c-9)/(14*c-9)
+2496144*a^11*b^13*x^(13*c-10)/(13*c-10)
+2704156*a^12*b^12*x^(12*c-11)/(12*c-11)
+2496144*a^13*b^11*x^(11*c-12)/(11*c-12)
+1961256*a^14*b^10*x^(10*c-13)/(10*c-13)
+1307504*a^15*b^9*x^(9*c-14)/(9*c-14)
+735471*a^16*b^8*x^(8*c-15)/(8*c-15)
+346104*a^17*b^7*x^(7*c-16)/(7*c-16)
+134596*a^18*b^6*x^(6*c-17)/(6*c-17)
+42504*a^19*b^5*x^(5*c-18)/(5*c-18)
+10626*a^20*b^4*x^(4*c-19)/(4*c-19)
+2024*a^21*b^3*x^(3*c-20)/(3*c-20)
+276*a^22*b^2*x^(2*c-21)/(2*c-21)
+24*a*b^23*x^(23*c)/(23*c)
+24*a^23*b*x^(c-22)/(c-22)-a^24/(23*x^23)
```