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I have run into an integral involving the complete elliptic integral, which can be put into the following form after changing integration variables to the modulus:

$$\int_0^{\sqrt{\frac{\alpha}{1+\beta}}} dk\, \frac{ k^{11} K(k) } {\sqrt{(\alpha-\beta k^2)^2 - k^4} (\alpha - \beta k^2)^{11/2}}$$

$K(k)$ is the complete elliptic integral of the first kind, where $k$ is the modulus. We can assume that $\alpha$ and $\beta$ are such the maximum value for $k$ is less than or equal to $1$. Are there any ways to get a closed form solution out of this? The indefinite integrals in G&R are not much help.

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+1 for remembering to mention your argument convention for elliptic integrals. It looks a bit gnarly as it stands, but I'll see what I can do. – J. M. Jul 21 '11 at 16:49

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