ISL 2010 A5

Denote by $\mathbb{Q}^{+}$the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the following equation for all $x, y \in \mathbb{Q}^{+}$: $$f\left(f(x)^{2} y\right)=x^{3} f(x y)$$ (Switzerland)