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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