ÖMO 2016 Problem 1

Let $\alpha\in\mathbb{Q}^+$ . Determine all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ that for all $x,y\in\mathbb{Q}^+$ satisfy the equation $$f\left(\frac{x}{y}+y\right) ~=~ \frac{f(x)}{f(y)}+f(y)+\alpha x.$$ Here $\mathbb{Q}^+$ denote the set of positive rational numbers.

(Proposed by Walther Janous)