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I trying to evaluate the following integral

$$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$

where the integration is w.r.t. $x$, and the other parameters are real positive.

Any idea?

FYI: with some change of variable, the following integral might be simpler to evaluate:

$$\int_0^\infty \dfrac {z^{\widetilde{m}-1} \Gamma(A,z)} {\left[ 1+(\tilde \eta z)^{\widetilde n} \right]^p} \,\mathrm dz$$

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  • $\begingroup$ By letting $t=\dfrac1{1+(ax)^n}$ , we have $\displaystyle\int_0^\infty\frac{x^{^{m-1}}}{(1+(ax)^n)^p}dx=\frac{B\big(p-\frac mn;\frac mn\big)}{n\,a^m}$ $\endgroup$ – Lucian Feb 8 '14 at 9:33
  • $\begingroup$ but did you consider the incomplete upper gamma function? please see my edit, that might help in someway. $\endgroup$ – kazekage Feb 8 '14 at 9:38

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