I encountered this integral in an answer of mine (here), but I wasn't able to find a closed form.
In detail, the integral is $$F_\gamma(x) = \int_0^{+\infty} \exp\left(-\frac{y\lambda}{a_1 - a_2y} - \mu h\right)\mathrm dh$$ where $$y = x - \frac{b_1 h}{b_2 h + 1}$$ and $\lambda, \mu, a_1, a_2, b_1, b_2 > 0$ with $a_1 + a_2 = b_1 + b_2 = 1$.
For context, the function $F_\gamma$ is the CDF of the random variable $\gamma$ (defined in the linked thread). From some simulations, it would seem that $\gamma$ behaves like a gamma distribution, so maybe this could be used to find a simpler form for $F_\gamma$ or its derivative $F'_\gamma(x)$ (to compare it to the PDF of the gamma distribution).
I was wondering if some progress can be made towards a simpler form for the random variable $\gamma$.