# 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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