No. 295 in my list of difficult integrals ...

Integral of the general power of : the object here is to establish the following reduction formula:

What I've tried: Integration by Parts, which gets me to:

... I waded further into this morass but I lost my boots.

I have also tried substitutions etc. but nothing useful happens, and trying to split it into a form with a is also unfruitful.

The usual: hints for a substitution or a splitting?