I have attached a file that defines a function to integrate the gamma_incomplete function and some forms involving it.
The pattern is
x^s*gamma_incomplete(e,a*x^t);
You can integrate it this way:
load("gm.mac")$
display2d:false;
->false
intgamma(x^s*gamma_incomplete(e,a*x^t),x);
-> gamma_incomplete(e,a*x^t)*x^(s+1)/(s+1)-gamma_incomplete((s+1)/t+e,a*x^t)/(a^((s+1)/t)*(s+1))
intgamma(%,x);
-> (gamma_incomplete(e,a*x^t)*x^(s+2)/(s+2)-gamma_incomplete((s+2)/t+e,a*x^t)/(a^((s+2)/t)*(s+2)))/(s+1)
-(gamma_incomplete((s+1)/t+e,a*x^t)*x-gamma_incomplete((s+1)/t+1/t+e,a*x^t)/a^(1/t))/(a^((s+1)/t)*(s+1))
intgamma(%,x);
-> (gamma_incomplete(e,a*x^t)*x^(s+3)/(s+3)-gamma_incomplete((s+3)/t+e,a*x^t)/(a^((s+3)/t)*(s+3)))/((s+1)*(s+2))
-(gamma_incomplete((s+1)/t+e,a*x^t)*x^2/2-gamma_incomplete((s+1)/t+2/t+e,a*x^t)/(2*a^(2/t)))/(a^((s+1)/t)*(s+1))
-(gamma_incomplete((s+2)/t+e,a*x^t)*x-gamma_incomplete((s+2)/t+1/t+e,a*x^t)/a^(1/t))/(a^((s+2)/t)*(s+1)*(s+2))
+a^(-(s+1)/t-1/t)*(gamma_incomplete((s+1)/t+1/t+e,a*x^t)*x-gamma_incomplete((s+1)/t+2/t+e,a*x^t)/a^(1/t))/(s+1)
That's all it can do. It does not work for complex number arguments.
Rich
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