What convex function should be used here?

Algebra Level 3

\[\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}. \]

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

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

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