Algebraic/Geometric Manipulations

Geometry Level 4

$\displaystyle{{ x }+{ y }={ \left( \cfrac { \sqrt { x } }{ \sqrt { y } } +\cfrac { \sqrt { y } }{ \sqrt { x } } \right) }^{ 2 }}$

$x$ and $y$ are positive real numbers satisfying the above equation.

Find Maximum value of $\displaystyle{\cfrac { 24\sqrt { x } +7\sqrt { y } +31\sqrt { xy } }{ \sqrt { xy } } }$

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