BALKAN-SL 2016 A1

Let $a, b, c$ be positive real numbers. Prove that

$$\sqrt{a^{3} b+a^{3} c}+\sqrt{b^{3} c+b^{3} a}+\sqrt{c^{3} a+c^{3} b} \geq \frac{4}{3}(a b+b c+c a)$$