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 $\ \alpha +\beta.$