Solution to the exponential integral $\int_{0}^{\infty}\exp\bigl(-pR-\frac{ER^{-f}}{B}\bigr)dR$

Question

I have a exponential integral $$\int\limits_{0}^{\infty}\exp\left(-pR-\frac{ER^{-f}}{B}\right)dR$$ where E, p, B, and f are constants, $E>0$

Context

This equation arrives when I try to consider how particles can randomly work from one location to another in arXiv powder bed. I can not find a solution to the integral, but I was told that the solution is $\exp\left( (-CE)^{\frac{1}{1+f}} \right)$. Could anyone help me how it comes up and what is C? Thank you.

Answer 1

Let's consider the case $E=0$. Then $$ \int_0^\infty \exp\big(-pR\big)\,dR = \frac{1}{p} $$ but the proposed "solution" is $\exp(0)=1$. Not a good sign for the case $E>0$.

Maple will do the case $f=1$: $$ \int_0^\infty \exp\left(-p R-\frac{ER^{-1}}{B}\right) = 2\sqrt{\frac{E}{Bp}} \mathrm{K}_1\left(2\sqrt{\frac{Ep}{B}}\right) $$ where $K_1$ is a Bessel function.