how to take integral of the Gamma Function

Question

Here is my question. I am trying to calculate the following integral:

$\int_{t}^{\infty} \Gamma(ax+b)dx$

I was not successful, although I tried many times.

I wonder if anyone can help me on this?

Thank you.

Answer 1

An indefinite integral of $$ \Gamma(s) := \int_0^\infty x^{s-1}e^{-x}\;dx,\quad \mathrm{Re}\;s > 0 , $$ is $$ F(s) := \int_0^\infty \frac{(x^{s-1}-1)e^{-x}}{\log x}\;dx,\quad \mathrm{Re}\;s > 0. $$ But $\lim_{s \to +\infty} F(s) = +\infty$, so $\int_t^{+\infty} \Gamma(ax+b)\;dx= +\infty$ if $a>1$.
Also $\lim_{s \to 0^+} F(s) = +\infty$, so $\int_t^{+\infty} \Gamma(ax+b)\;dx= +\infty$ if $0 < a < 1$.