It's not elegant but ...
differentiating num w.r.t. x gives:
Now provided num is real and non zero the original integral is
I'm working my way through a series of ever tougher integrals. I'm stuck at no. 283:
... and there are some further even tougher ones.
I understand that it is supposed to evaluate to:
depending on the sign of
I have already tried:
a) completing the square on the expression under the square root
b) substituting and
c) integrating by parts with and etc.
I've also tried working backward, differentiating the expression, to see where it gets me, but the penny is not dropping.
The good cause this is for is ProofWiki which now has over 10000 proofs up.