Triple Inequalities

Algebra Level 5

Given positive real numbers \( (x,y,z) \) such that \( xy + yz + xz = 5\). The minimum value of \[ \dfrac{x^2}{y^3} + \dfrac{y^2}{z^3} + \dfrac{z^2}{x^3} + x + y + z \]

is \(
\frac{ a \sqrt{b} } { c} \), where \(a, b, \) and \(c\) are positive integers, \( \gcd(a,c) = 1 \) , and \(b\) is square free. What is the value of \( a + b + c \)?


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