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Calculus Level 5

If

\[\large{u=\csc ^{ -1 }{ { \left( \sqrt { \frac { \sqrt { x } +\sqrt { y } }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ y } } } \right) } } }\]

Then

\(\large{x^2 \frac{\partial^{2} u}{\partial x^2}+2xy \frac{\partial^{2} u}{\partial x \partial y}+y^2 \frac{\partial^{2} u}{\partial y^{2}}=\frac{\tan u}{A} \left(B+\tan^{C} u \right)}\)

where \(A,B,C\) are positive integers.

Find \(A+B+C\)

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