I am trying to find the following integral
\begin{equation} \int_{0}^{\infty}\left(1+\kappa\text{e}^{\left(b-c\right)t}\right)^{-\frac{b}{b-c}}e^{-at}dt \end{equation}
Intuitively, I have started to try hypergeometric function but could not go further. I have tried on Mathematica and saw that there is a solution. However, I would like to understand how it can be solved analytically. I appreciate any solution or hints.
I am also interested in the indefinite form of the integral.