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 $x+y$ .