A not so easy differential equation

Calculus Level 5

\[\dfrac { dy }{ dx } \cdot \dfrac { { d }^{ 3 }y }{ d{ x }^{ 3 } } =3{ \left( \dfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) }^{ 2 }\] The general solution of the differential equation above is of the form \({ y }^{ \alpha }=\dfrac { \sqrt [ \beta ]{ kx+f } }{ g } +c\), where \(k,\) \(f,\) \(g,\) and \(c\) are arbitrary constants and \(\alpha\) and \(\beta\) are coprime positive integers.

Find the value of \(\ \alpha +\beta.\)

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