All these roots

Algebra Level 5

\[\large \sqrt{\frac{x}{y+z}}+\sqrt{\frac{y}{x+z}}+\sqrt{\dfrac{z}{x+y}}+2\sqrt{\dfrac{2(x^2+y^2+z^2)}{xy+yz+xz}}\]

Let \(x,y\) and \(z\) be positive reals. If the minimum value of the expression above can be expressed in the form \(\dfrac{\alpha\sqrt{\beta}}{\gamma}\), where \(\alpha,\beta\) and \(\gamma\) are positive integers with \(\beta\) square-free and \(\alpha\) and \(\gamma\) coprime, find \(\alpha +\beta +\gamma\).

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