# Which inequality should I use?

Algebra Level 4

$\dfrac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\dfrac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\dfrac{c^2}{\sqrt{(1+c^3)(1+a^3)}}$

Let $$a, b, c$$ be positive real numbers such that $$abc=8$$ . and if the minimum value of the above expression can be expressed as $$\frac m n$$, where $$m$$ and $$n$$ are coprime positive integers, find the value of $$m+n$$.

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