Integral Calculus: JEE Main 2024 Mathematics question

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 $\pi / 6$ and $\pi / 4$, respectively with positive $x$-axis. If 27 \int_\limits1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3} where $\alpha, \beta$ are integers, then the value of $\alpha+\beta$ equals