Sets, Relations and Functions: JEE Main 2023 Mathematics question

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