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 $g(x)=x-[x]$ (where $[x]$ is the greatest integer function) is $(\alpha,\beta)$, then $\alpha^{2}+\dfrac{5}{\beta}$ is equal to: