Let the function,
be differentiable for all \( x\in \mathbb{R} \), where \( a>1, b\in \mathbb{R} \). If the area of the region enclosed by \( y=f(x) \) and the line \( y=-20 \) is \( \alpha+\beta\sqrt{3} \), \( \alpha,\beta\in \mathbb{Z} \), then the value of \( \alpha+\beta \) is ______.
