Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1,f(1))$ and $(3,f(3))$ make angles $\dfrac{\pi}{6}$ and $\dfrac{\pi}{4}$ respectively with the positive $x$-axis. If $27\displaystyle\int_{1}^{3}\big((f'(t))^{2}+1\big)f'''(t),dt=\alpha+\beta\sqrt{3}$, where $\alpha,\beta$ are integers, then the value of $\alpha+\beta$ equals: