An algebra problem by A Former Brilliant Member

Algebra Level 4

If $\alpha$ , $\beta$ and $\gamma$ are roots of equation $x^3+4x+1$ .Then, find $\left(\dfrac{1}{\alpha+\beta}+\dfrac{\alpha+\beta}{\gamma^2}\right)+\left(\dfrac{1}{\beta+\gamma}+\dfrac{\beta+\gamma}{\alpha^2}\right)+\left(\dfrac{1}{\gamma+\alpha}+\dfrac{\gamma+\alpha}{\beta^2}\right)$

Inspiration.