Primitive of the following integral This is my first post, so feel free to tell me anything I'm doing wrong:
I'm trying to find if there is a primitive or any know solution to the following integral:
$$t = \int \frac{\rho d\rho}{\sqrt{-a+b\rho^2-c\rho^3}}$$
I suppose it will be (if there is one) something related with the elliptic integrals, because it has to do with the movement of a particle inside an inverted cone (Hooke's inverted cone) but I can't figure if this is true. Thank you so much in advance!
 A: For this kind of problem, a good idea is to factorize the polynomial$$I = \int \frac{\rho \,d\rho}{\sqrt{-a+b\rho^2-c\rho^3}}=\int\frac{\rho \,d\rho}{\sqrt{-c (\rho -\alpha ) (\rho -\beta ) (\rho -\gamma )}}$$ and, using a CAS,
$$J=\frac{\sqrt{\beta -\alpha } \sqrt{-c (\rho-\alpha ) (\rho -\beta ) (\rho -\gamma )}}{2(\rho-\alpha )^2} \,I$$ is given by
$$J=\frac{\sqrt{\beta -\alpha } (\rho-\beta  ) (\rho-\gamma  )}{(\rho-\alpha )^2}-\frac{ \sqrt{\frac{\rho -\beta}{\rho-\alpha }} \sqrt{\frac{\rho-\gamma
   }{\rho-\alpha }}}{\sqrt{\rho-\alpha}}\, \Delta$$ where
$$\Delta=\beta  F\left(i \sinh ^{-1}\left(\frac{\sqrt{\beta -\alpha }}{\sqrt{\alpha
   -\rho }}\right)|\frac{\alpha -\gamma }{\alpha -\beta }\right)+(\alpha
   -\beta ) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\beta -\alpha
   }}{\sqrt{\alpha -\rho }}\right)|\frac{\alpha -\gamma }{\alpha -\beta
   }\right)$$ where appear the elliptic integrals of the first and second kind.
A: Please consider this as a comment, rather than an answer to your question. A quick look using Mathematica gave this (click on the image to enlarge):

