Školjka

%V0 Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ $$f\left( f(x)^2y \right) = x^3 f(xy).$$ Proposed by Thomas Huber, Switzerland