Which inequality should I use?

Algebra Level 4

a2(1+a3)(1+b3)+b2(1+b3)(1+c3)+c2(1+c3)(1+a3) \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,ca, b, c be positive real numbers such that abc=8abc=8 . and if the minimum value of the above expression can be expressed as mn \frac m n , where mm and nn are coprime positive integers, find the value of m+nm+n.

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