I might have an angle on this:
a) Complete the square to get the integral in the form
b) Substitute to get it in the form
c) Use the standard result
but it's late and I need to sleep on it.
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?