Let $D$ 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 $g_i D \rightarrow R$ defined by $g(x)=x-[x],\left([x]\right.$ is the greatest integer function), is $(\alpha, \beta)$ then $\alpha^2+\frac{5}{\beta}$ is equal to-