Apply Cauchy-Schwarz?

Algebra Level 5

$\large \dfrac {a}{b(b+c)^2} + \dfrac {b}{c(c+a)^2} + \dfrac {c}{a(a+b)^2}$

Let $a$ , $b$ and $c$ be positive reals such that $ab+bc+ca=k$ , where $k$ is a positive real as well.

Let the minimum value of the above expression be $\dfrac {x}{y} k^{z}$ , where $x$ and $y$ are coprime positive integers and $z$ is an integer. Find $x+y+z$ .

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