An Integral from Hardy
Here's an integral from G. H. Hardy:
$$
\int \dfrac{dx}{\sqrt{x+a} + \sqrt{x+b}}
$$
It doesn't fit any of the standard undergraduate forms. There's no obvious \(u\)-substitution, integration by parts doesn't help, and a trigonometric substitution turns it into a mess.
The Setup
The trouble is the sum of two square roots in the denominator. With just one square root the integral is a power-rule exercise:
$$
\int \dfrac{dx}{\sqrt{x+a}} = 2\sqrt{x+a} + C
$$
But the sum resists the usual substitutions. Its derivative involves both roots in a form that …