$f(yf^2(x))=x^3f(xy)$ for all $x,y \in {\mathbb{Q}}^{+}$

Question

Find all functions $f: {\mathbb{Q}}^{+} \rightarrow {\mathbb{Q}}^{+}$ such that:
$f(yf^2(x))=x^3f(xy)$ for all $x,y \in {\mathbb{Q}}^{+}$
My progress:
$P(x;1) \Rightarrow f(f^2(x))=x^3f(x)$
Easily we have $f$ is injective.
$P(1;y) \Rightarrow f(yf^2(1))=f(y) \Rightarrow f(1)=1$
$P \left(x;\dfrac{1}{x} \right) \Rightarrow f \left(\dfrac{f^2(x)}{x} \right)=x^3$
$\Rightarrow f \left(\dfrac{f^2(x)}{x} \right) \cdot f(x)=f(f^2(x))$
I wondered if f is multiplicative.