Help needed with derived integral

Hi

I reeeaallly need some help here. This equation

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(listed in numerous journals) is related to the calculation of the capacity factor of a wind turbine from certain parameters. I have them all (weibull k & c values, turbine characteristics etc), so should be able to do it. However, there are some gamma functions that i have absolutely no idea what to do with. As you can tell - calculus is not my strong point.

Could someone help with this? It'd be hugely appreciated.