I am working with the following problem associated with the overall capacity of a Rayleigh fading channel: \begin{equation}\int_0^\infty {\frac{{cy{e^{by}}{\Gamma}\left[ {0,\frac{y}{b}} \right]}}{{1 + ay}}} dy\end{equation} I tried to find a closed form solution of the integral using different numerical approaches. I also tried to use Wolfram Mathematica 12.3. However, I couldn't come to a satisfying solution.
Then, I have tried to solve the integral manually using integration by parts assuming ${\frac{{cy{e^{by}}}}{{1 + ay}}}$ as $u$, and ${\Gamma}\left[ {0,\frac{y}{b}} \right]dy$ as $dv$; and found a closed form solution. I am not sure the approach is a correct way to solve an integral.
Could you please let me know your thoughts and help me to solve the integral? Thank you very much.