An Integral from Hardy
Every once in a while you run across an integral that looks straightforward at first glance, but the more you stare at it, the less obvious it becomes how to proceed. This is one of those integrals. It's the sort of problem that G. H. Hardy - the great English mathematician who taught at Cambridge for decades and famously collaborated with Ramanujan - liked to pose to students as a way to sharpen their instincts about integration.
Here's the integral:
$$
\int \dfrac{dx}{\sqrt{x+a} + \sqrt{x+b}}
$$
At a glance, this doesn't fit any of the …