9.2.5 Integrals depending on a parameter

Problem: Applying differentiation with respect to the parameter, calculate the following integral: \[ I(\alpha, \beta)=\int_{0}^{+\infty} \frac{\ln \left(1+\alpha^{2} x^{2}\right) \tan ^{-1}(\beta x)}{x^{3}} d x, \quad \alpha, \beta>0 . \]