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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