OBM 2013 Problem 3

Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^*$ from the non-zero reals to the non-zero reals, such that $$f(x+y) \left(f(x) + f(y)\right) = f(xy)$$ for all non-zero reals $x, y$ such that $x+y \neq 0$ .