Let $\mathbb{R}^+$ be the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying the equation
$$xf(x^2)f(f(y)) + f(yf(x)) = f(xy)(f(f(x^2)) + f(f(y^2))).$$
for all $x, y \in \mathbb{R}^+$.