JBMO-SL 2008 A7

Let $a, b$ and $c$ be a positive real numbers such that $a b c=1$. Prove the inequality

$$\left(a b+b c+\frac{1}{c a}\right)\left(b c+c a+\frac{1}{a b}\right)\left(c a+a b+\frac{1}{b c}\right) \geq(1+2 a)(1+2 b)(1+2 c)$$