Definite Integral involving exponentials and arbitrary powers

Question

I have the following integral:

$$\int_0^{+\infty} t^{z-1} e^{-t} \frac1{(k_1t + 1)^s}\frac1{(k_2t + 1)^s}\mathrm dt$$

where $k_1>0$, $k_2>0$ and $s>0$. How would you suggest to solve it?

Answer 1

Knowing that: \begin{align}{\int_0^{\infty}t^{z - 1}\exp{-t}\,% {1 \over ({kt + 1})^{s}}\,dt} &={\exp({1 \over 2k})k^{({1 - z - s})/2}\ \Gamma({z}){\rm W}_{({1 - z -s})/2,\ ({z - s})/2}({1 \over k})} \end{align} where W is the .