Školjka

%V0 Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca \leqslant 3abc$. Prove that $$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3 \leqslant \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right) \text{.}$$ Proposed by Dzianis Pirshtuk, Belarus