Definite integral involving $\exp((-1/x) - x)$

Question

I have a small integration problem. Just for context, this comes out of a bayesian exercise involving gamma functions.

$$\int_{0}^{\infty} x^\alpha e^{-\left( \beta x^{-1} + \gamma x \right)} dx$$

where $\beta>0, \gamma >0$.

I can't get my head around what the right change of variables is. I have tried, naively:

$$y = \beta x^{-1} + \gamma x$$

but I got stuck pretty quickly. Any idea what the best change of variables would be?

Thanks a lot in advance! :)

Answer 1

I am sure a good MSEer will provide a derivation, which certainly will be instructive, but in the meantime: \begin{equation} \int_0^\infty\,x^{\alpha - 1}\exp(-px - q/x)\, dx = 2(q/p)^{\alpha/2}K_\alpha(2\sqrt{pq}). \end{equation} 2.3.16.1, Prudnikov, Brychkov, Marichev, Integrals and Series, vol 1, Elementary Functions; easier access at DLMF 10.32.10.