How to integrate $\frac{x^N\log(1+x)}{\sqrt{x^2+x_1^2}\sqrt{x^2+x_2^2}}$?

Question

I am trying to compute the integral $$\int_{x_0}^{1}\frac{x^N\log(1+x)}{\sqrt{x^2+x_1^2}\sqrt{x^2+x_2^2}}\text{d}x$$ where $x_0, x_1$ and $x_2$ are related to some parameters $\kappa_\pm$ by $$x_0=\sqrt{\frac{\kappa_-}{\kappa_+}},x_1=\sqrt{\frac{1+\kappa_-}{1+\kappa_+}},x_2=\sqrt{\frac{1-\kappa_-}{1-\kappa_+}}$$ and $N$ is in $\{-1,0,1,2\}$. I expect the result to be some combination of logs and dilogs in the $x_i$ variables. However, I don't seem to be able to compute it. If the denominator was instead the square-root of a second-order polynomial, the integral would be computable with exactly the expected behaviour. However, in the present form it appears to be much harder.