Let \( D \) be the domain of the function \( f(x) = \sin^{-1}\!\left( \log_{3x} \left( \dfrac{6 + 2\log_3{x}}{-5x} \right) \right) \). If the range of the function \( g : D \to \mathbb{R} \) defined by \( g(x) = x - [x ] \), (\( [ x ] \) is the greatest integer function), is \( (\alpha, \beta) \), then \( \alpha^{2} + \dfrac{5}{\beta} \) is equal to:
