I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below),

$$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\frac{1}{2},1-\frac{1}{q-1},\frac{3}{2},1-\left(\frac br\right)^{1-q}\right)\right)$$

where $b$ is just some positive constant, while $-\infty<q<1$.

Basically I want to series expand $\rho$ for large $r$ (i.e. as $r\to \infty$) and then invert the series to obtain $r(\rho)$. I have tried some readily available asymptotic expansion of ${}_2F_1$.

Basically I got,

$$\rho\sim r\sqrt{1-(b/r)^{1-q}}$$

as the leading term in the expansion. I am not that sure what to pick as the next two correction terms from the link given the allowed values $-\infty<q<1$. Is there a detailed easy way of finding the next two correction terms in the expansion, given the allowed values of $q$? Thank you.