Help needed with Elliptic Integrals

[TerryW]

Homework Statement

See Attempt at a Solution

Relevant Equations

See Attempt at a Solution

I want to find the solution to the integral

$\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}$

I can see that $\frac{d^2u}{d\theta^2} = A +Bu+Cu^2$ is a Weierstrass elliptic function, which can be generated from $\Large(\normalsize\frac{du}{d\theta}\Large)\normalsize^2 = c-u^2 +2u^3$ (A = 0, B=-1, C=3)

So does this make my integral an elliptic integral? I haven't been able to find a table of integrals anywhere which contains an integral of this form so I'm a bit stuck.

TerryW

 

[renormalize]

I want to find the solution to the integral

$\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}$

Mathematica can evaluate the indefinite integral on the right side:

Here, $\text{EllipticF}\left[\phi,m\right]$ is the elliptic integral of the 1rst kind $F\left(\phi\mid m\right)$ and the $r_k$ are the 3 roots of the cubic polynomial equation $c-r^2+2\,r^3=0$. According to the documentation, the roots are indexed as follows:
"The root indexing representation $\text{Root}\left[f,k\right]$ applies to polynomial functions $f$ only. The indexing of roots takes the real roots first, in increasing order. For polynomials with rational coefficients, the complex conjugate pairs of roots have consecutive indices."