I recently came across the $$\int{w^a(1-w)^b\mathrm d w},$$ which looked ridiculously simple at first, but I subsequently discovered that I could not reduce it to some elementary form.
To clarify, I do not mean the following cases: when $a$ or $b$ is a nonnegative integer, for then integrating by parts does it; when $a$ and $b$ are negative integers, in theory one could do this since we only have a rational function, especially if $|a|$ and $|b|$ are not too large.
Thus, I mean that case when both $a$ and $b$ are not nonnegative integers. One might on impulse think that setting $a=b$ might simplify matters, but even in this restricted case I couldn't do anything -- integrating by parts is out of the question since it leads to an infinite series; changing the integrand in several ways doesn't do anything interesting; even differentiating with respect to $a=b$ only complicates the matter. I then began to wonder if it was possible at all.
So my question is this:
Is it possible to reduce the above integral to elementary form? If so, what is a (theoretical) way to go about it? If not, why not? (I was originally interested in the case where $a\ne b,$ but the opposite case too should throw some light on the matter; in any case, progress will be appreciated.)