BALTIC-WAY 2021 Problem 1

Let $n$ be a positive integer. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the equation

$$(f(x))^{n} f(x+y)=(f(x))^{n+1}+x^{n} f(y)$$

for all $x, y \in \mathbb{R}$.