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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.

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    $\begingroup$ What is the background of this question? What have you tried? I converted your picture into latex form; please check that I didn't make a mistake. $\endgroup$ – Eckhard Dec 25 '13 at 21:51
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    $\begingroup$ Did you mean $\alpha + \beta x$ instead of $\alpha+\beta \gamma$ in the denominator of the integrand? $\endgroup$ – Sasha Dec 25 '13 at 22:19
  • $\begingroup$ @Eckhard: this is related to comm. channels. I tried change of variable and assumed z=bsqrt(alpha+betax), but the integral got worse. $\endgroup$ – kazekage Dec 26 '13 at 5:28
  • $\begingroup$ @Sasha: Yes, thanks for the correction. $\endgroup$ – kazekage Dec 26 '13 at 5:29
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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.

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