Problem: Let $a,b,c>0: \sqrt{a^4+8a}+\sqrt{b^4+8b}+\sqrt{c^4+8c}=3(a^2+b^2+c^2).$ Prove that: $$\frac{a^4+2a+b(2a^2+b)}{a^2(b^2+2a)}+\frac{b^4+2b+c(2b^2+c)}{b^2(c^2+2b)}+\frac{c^4+2c+a(2c^2+a)}{c^2(a^2+2c)}\ge 6.$$
My attempt: The problem is equivalent to: $$\sum_{cyc}{\left(\frac{1}{a^2}+\frac{a^2+2b}{b^2+2a}\right)}\ge6$$ It is complicated problem at first glance. I think the only ostable is using condition. Anyone help me end the rest part? Thanks!