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