I'm looking at the following integral
$t = \int_{x_0}^{x(t)} \frac{dx'} {\sqrt{A+Bx'+Cx'^2+Dx'^3+Ex'^4+Fx'^6}}$
and want to know what the inverse function $x(t)$ is, knowing $x(0)=x_0$ and $A=-Bx_0-Cx_0^2-Dx_0^3-Ex_0^4-Fx_0^6$. I found $x(t)$ numerically and it looks like a periodic function.
This arises from trying to solve for $x$ in a system where $\frac{d^2}{dt^2}x$ is a 5th degree polynomial of $x$ (with no 4th degree term in this case).
I know that if the polynomial under the square root was 3rd or 4th degree, $t$ would be an elliptic integral and $x(t)$ could be related through some transformation to elliptic functions (see this stack exchange). Is there an analogue to elliptic functions when the polynomial under the root is of degree 6 ?
