I would like to evaluate the following integral related to the exponential distribution. Let $\delta>1$, and $0<p<1$ and $0<\epsilon<1/\delta$ be reals. We have that
$$ \int_0^{\epsilon p}\exp\left(-x\cdot\tfrac{1}{p}\right)\,\,\mathrm{d}x = p(1-e^{-\epsilon}) $$
I am interested in calculating the related integral
$$ \int_0^{\epsilon p}\exp\left(-x\cdot\tfrac{1-\delta x}{p-\delta x}\right)\,\mathrm{d}x. $$
I have tried Mathematica to no avail. I would appreciate any help. I would love to hear the reasoning behind the steps if someone has a solution.
Thanks a lot!