I am trying to calculate the following integral: $$ \int_0^{+\infty} \frac{t^{a-1}e^{-bt}}{(d + et)^r} \mathrm{erfc}(\sqrt{wt}) dt = \int_0^{+\infty} \frac{t^{a-1}e^{-bt}}{(d + et)^r} \Gamma(1/2,wt) dt, $$ where all parameters are positive. As you can see, it is evolving the incomplete gamma fcuntion/error function. I tried to look in integrals/series books to find an appropriate closed-form expression without success.
I also tried to find a series representation for the incomplete gamma function for half-integers, but no success.
I would like to ask if any solution exists ?
Thank you.