Mathematics Question from JEE Main 2024 · 30 January, Shift 2

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 the positive $x$-axis. If $27\int_{1}^{3}\Big(\big(f'(t)\big)^2 + 1\Big) f''(t)\, dt = \alpha + \beta\sqrt{3}$ where $\alpha$ and $\beta$ are integers, then the value of $\alpha + \beta$ equals: