I found the following formula in "INTEGRALS AND SERIES, vol.3" by Prudnikov, Brychkov and Marichev (page 188, eq.5).
$$\int_0^{\infty} \frac{x^{\alpha-1}}{\sqrt{(a+x)^2+z^2}}K(\frac{2\sqrt{ax}}{\sqrt{(a+x)^2+z^2}})dx = \frac{\sqrt{\pi}}{4}\Gamma(\frac{a}{2})\Gamma(\frac{1-a}{2})(a^2+z^2)^{(\alpha-1)/2}P_{-\alpha}(\frac{z}{\sqrt{a^2+z^2}})$$$$\int_0^{\infty} \frac{x^{\alpha-1}}{\sqrt{(a+x)^2+z^2}}K(\frac{2\sqrt{ax}}{\sqrt{(a+x)^2+z^2}})dx = \frac{\sqrt{\pi}}{4}\Gamma(\frac{\alpha}{2})\Gamma(\frac{1-\alpha}{2})(a^2+z^2)^{(\alpha-1)/2}P_{-\alpha}(\frac{z}{\sqrt{a^2+z^2}})$$
$K$: complete elliptic integral of the first kind.
$\Gamma$: Gamma function
$P$: Legendre function of the first kind.
$\alpha$: complex value
The book says this is valid for $a>0, Re\{z\}>0, 0 < Re\{\alpha\}<1$. Could anybody tell me how this can be derived ?
I think this formula would be useful for evaluating Mellin transform of a function that contains the complete elliptic integrals.
Thank you in advance.