JBMO-SL 2018 A6

Let $a, b, c$ be positive numbers such that $a b+b c+c a=3$. Prove that

$$\frac{a}{\sqrt{a^{3}+5}}+\frac{b}{\sqrt{b^{3}+5}}+\frac{c}{\sqrt{c^{3}+5}} \leq \frac{\sqrt{6}}{2}$$