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.