I need some help on a possible way to integrate this:
$$ \int_0^\infty{x^{m-1}\mathrm{e}^{-\lambda x}\left[\frac{\operatorname{K}_\nu\left(b\sqrt{\alpha+\beta x}\right)}{\left(b\sqrt{\alpha+\beta x}\right)^\nu}\right]}\mathrm{d}x $$ where $m$, $\lambda$, $b$, $\alpha$, $\beta$ are constants and $\operatorname{K}_\nu(\cdot)$ is the modified Bessel function of the second type.
Maple has this special case in terms of the Meijer G function: $$ \int _{0}^{1}\!{x}^{m-1}{{\rm K}_\nu\left(b\sqrt {\beta\,x}\right)} \left( b\sqrt {\beta\,x} \right) ^{-\nu}{dx}={2}^{-3+2\,m-\nu} \left( b\sqrt {\beta} \right) ^{-2\,m}{b}^{2}\beta\, G^{2, 1}_{1, 3}\left(\frac{\beta{b}^{2}}{4}\, \Big\vert\,^{0}_{m-1-\nu, m-1, -1}\right) $$ assuming $\nu$ is a positive integer.