Find all functions $f:\mathbb{R}_{>0}\rightarrow \mathbb{R}_{>0}$ such that $$ f(xy+f(x))=f(f(x)f(y))+x, \forall x,y\in \mathbb{R}_{>0} $$
If someone can give me a hint for this question I would be very grateful.
edit: We can see that $f$ is injective because: $f(x)=f(y)$ then the equation above tell's that $x=y$.