I have been trying to invert the hypergeometric function below. Unfortunately, it cannot be inverted exactly so I tried feeding up the function with random values of $ q$ . Luckily, there are two values where this function can be inverted exactly, that is $ q=-1$ and $ q=1/3$ . Is there an effective way or routine to find $ q$ values in which the function has an exact inverse?

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

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