Inequality. $(a^2+bc)(b^2+ca)(c^2+ab) \geq abc(a+b)(b+c)(c+a)$ Let $a,b,c$ be three real positive(strictly) numbers. Prove that: 
$$(a^2+bc)(b^2+ca)(c^2+ab) \geq abc(a+b)(b+c)(c+a).$$
I tried : 
$$abc\left(a+\frac{bc}{a}\right)\left(b+\frac{ca}{b}\right)\left(c+\frac{ab}{c}\right)\geq abc(a+b)(b+c)(c+a) $$
and now I want to try to prove that for example $$a+\frac{bc}{a} \geq a+b$$
but I don't know if is is a good idea. 
Thanks:) 

By C-S $$\sum_{cyc}(a^2+bc)=\prod_{cyc}\sqrt{(a^2+bc)(ac+b^2)}\geq\prod_{cyc}\left(\sqrt{a^3c}+\sqrt{b^3c}\right)=abc\prod_{cyc}(a+b).$$
Done!
