I want to integrate a function that envolves a root of a non polynomial, to be more specific, this is the function:
$$ x(u) = \int \frac{du}{\sqrt{\left( 1+ \frac{(c-1)M}{2-3Mu}\right) \left(2Mu^{3}-u^{2} + \frac{1}{D^{2}} \right)}} \,\, , $$
where $c,M, D$ are all real parameters. If I didn't had the first term inside the root I would know to to integrate, but with this term all gets more difficult, so I don't have any insight about how to perform this integral analytically, I'm avoiding numerical methods. Thanks in advance!