While I was elaborating on a fluid mechanics problem, I came across improper (infinite) integrals of the form $$ I_{m,n} (\alpha, \rho) = \int_0^\infty \frac{q^n e^{-q}}{1+\alpha q} J_m(\rho q) \mathrm{d} q, $$ wherein $n \in \{1,2\}$ and $m \in \{0, 1, 2\}$. In addition, $\alpha \ge 0$ and $\rho \ge 0$.
For $\alpha = 0$, those integrals can easily be evaluated. I was wondering whether there exists an approach to tackle those integrals when $\alpha \ne 0$. Any hints or advice is highly appreciated.
Thank you!