Find the value of the determinant given below, without expanding it at any stage.

Find the value of the determinant given below, without expanding it at any stage. $\begin{vmatrix} \beta\gamma & 1 & \alpha(\beta+\gamma) \\ \gamma\alpha & 1 & \beta(\gamma+\alpha) \\ \alpha\beta & 1 & \gamma(\alpha+\beta) \end{vmatrix}$