What convex function should be used here?

Algebra Level 3

1a3+1+1b3+1+1c3+1.\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}.

Positive reals a, b, ca,\ b,\ c satisfy 1a+1b+1c=3.\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3.

If the maximum value of the above expression is in the form xy\dfrac{x}{\sqrt y}, where xx and yy are integers with yy square-free, find x+yx+y.

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