HMIC 2023 Problem 1

Let $\mathbb{Q}^{+}$denote the set of positive rational numbers. Find, with proof, all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ such that, for all positive rational numbers $x$ and $y$, we have

$$f(x)=f(x+y)+f\left(x+x^{2} f(y)\right)$$