Hi again,
Sorry for another post on this. I figured out the problem with the rules and I also found a bug in quad_qagi, I think
but it may just be overflow. Any here it is.
(%i43) display2d:false;
(out43) false
(%i44) out4;
(out44) -(gamma_incomplete(6/7,9*x^7)*x-gamma_incomplete(1,9*x^7)/9^(1/7))/(7*9^(6/7))
(%i45) quad_qagi(%,x,0,inf);
gamma_incomplete: continued fractions failed for gamma_incomplete(1.0, 4.3682654441477153E+19).
(out45) quad_qagi(-(gamma_incomplete(6/7,9*x^7)*x-gamma_incomplete(1,9*x^7)/9^(1/7))/(7*9^(6/7)),x,0,inf,epsrel =
1.0E-8,epsabs = 0.0,limit = 200)
(%i46)
Now for the rules that work.
matchdeclare([a,s],freeof(x),x,true,[m,n],nonnegintegerp)$
block(
[simp:false],
tellsimp('integrate(gamma_incomplete(s, a*x^n), x),x*gamma_incomplete(s, a*x^n) - (x*gamma_incomplete(1/n + s,
a*x^n))/(a^(1/n)*(x^n)^(1/n))),
tellsimp('integrate(gamma_incomplete(s, a*x^n)*x^m, x),x*gamma_incomplete(s, a*x^n)*x^m/(m+1)
- (x*gamma_incomplete((m+1)/n + s, a*x^n))/((m+1)*a^((m+1)/n)*(x^n)^(1/n)))
)$
remove([a,s,x,m,n],matchdeclare)$
declare(integrate,linear);
assume(x>0);
/* eof */
Rich