I'm interested in computing the integral $$I(x,y)=\frac12\int_0^x \frac{dt}{(t-y)\sqrt{t(t-x)(t-1)}}$$ If not for the additional pole that would be a complete elliptic integral of the first kind $$\frac12\int_0^x \frac{dt}{\sqrt{t(t-x)(t-1)}}=\int_0^1\frac{dt}{\sqrt{(1-t^2)(1-xt^2)}}=:K(x)$$ I know that by a general theory my integral should be expressible in terms of the three basic elliptic integrals, but I can not find a good way to do it. In fact, I can not even prove the previous identity (with no additional pole). I'm interested in the final result but also in the method. If someone can recommend an accessible introduction to elliptic integrals and their manipulation that would be very helpful.
Ok, so the second formula is obtained by rescaling $t\to xt$ and then changing $t\to t^2$. The same procedure converts initial integral to the elliptic integral of the third Kind in a more or less canonical form. I guess the technical part of the question is gone, but I'm still interested in useful references.
