BALTIC-WAY 2014 Problem 3

Positive real numbers $a, b, c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove the inequality

$$\frac{1}{\sqrt{a^{3}+b}}+\frac{1}{\sqrt{b^{3}+c}}+\frac{1}{\sqrt{c^{3}+a}} \leqslant \frac{3}{\sqrt{2}} .$$