Let $y = y ( x )$ be the solution of the differential equation $( x + y + 2 ) ^ { 2 } d x = d y , y ( 0 ) = - 2$. Let the maximum and minimum values of the function $y = y ( x )$ in $\left[ 0 , \frac { \pi } { 3 } \right]$ be $\alpha$ and $\beta$, respectively. If $( 3 \alpha + \pi ) ^ { 2 } + \beta ^ { 2 } = \gamma + \delta \sqrt { 3 } , \gamma , \delta \in \mathbb { Z }$, then $\gamma + \delta$ equals $\_\_\_\_$