CZ-SK-MO 2014 Problem 6

For arbitrary non-negative numbers $a$ and $b$ prove inequality $\frac{a}{\sqrt{b^2+1}}+\frac{b}{\sqrt{a^2+1}}\ge\frac{a+b}{\sqrt{ab+1}}$ , and find, where equality occurs.

(Day 2, 6th problem authors: Tomáš Jurík, Jaromír Šimša)