IN-TST 2010 Problem 4

Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$ . Prove that $$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})$$