Apply Cauchy-Schwarz?

Algebra Level 5

ab(b+c)2+bc(c+a)2+ca(a+b)2\large \dfrac {a}{b(b+c)^2} + \dfrac {b}{c(c+a)^2} + \dfrac {c}{a(a+b)^2}

Let aa, bb and cc be positive reals such that ab+bc+ca=kab+bc+ca=k, where kk is a positive real as well.

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

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