Definite Integration

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