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 π / 6 and π /…

Let $\displaystyle y=f(x)$ be a thrice differentiable function in $\displaystyle (-5,5)$. Let the tangents to the curve $\displaystyle y=f(x)$ at $\displaystyle (1, f(1))$ and $\displaystyle (3, f(3))$ make angles $\displaystyle \pi / 6$ and $\displaystyle \pi / 4$, respectively with positive $\displaystyle x$-axis. If $\displaystyle 27 \int_1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3}$ where $\displaystyle \alpha, \beta$ are integers, then the value of $\displaystyle \alpha+\beta$ equals