ELMO SL 2014 A5

Let $\mathbb R^\ast$ denote the set of nonzero reals. Find all functions $f: \mathbb R^\ast \to \mathbb R^\ast$ satisfying $$f(x^2+y)+1=f(x^2+1)+\frac{f(xy)}{f(x)}$$ for all $x,y \in \mathbb R^\ast$ with $x^2+y\neq 0$ .

Proposed by Ryan Alweiss