# Finding the inverse of an incomplete beta function

Is there a rigorous way of inverting

$$\rho(r)=\frac{b_{0}}{1-q}B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$$

where $$B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$$ is an incomplete beta function, $$b_{0}$$ is some positive constant, $$-\infty. Basically I want to obtain $$r(\rho)$$. I am thinking of getting the asymptotic expansion of $$\rho(r)$$ (since I am interested in the regime where $$r$$ is very large) and then invert using Lagrange inversion method. Maybe I only need the first three terms of the expansion. Am I allowed to do this? Or is there a more convenient way of inverting the given function containing an incomplete beta function?