I originally saw this logarithmic integral pop up on the Integrals and Series forum but due to the inactivity there no conversation has developed.
I originally tried generalizing the integral to
$$I(\alpha,\;\beta)=\int_0^1\frac{\ln(x)\ln(\alpha x+1)\ln(\beta(x^2+x)+1)}{(1-x)(1+x^2)}dx$$
and differentiating under the integral sign with respect to $\alpha$ and $\beta$ but it didn't seem to go anywhere. However, I did get the related integrals
$$\int_0^1\frac{(x^2+x)\ln(x)}{(1-x)(1+x^2)(x^2+x+1)}dx=-\frac{17\pi^2}{432}$$
and
$$\int_0^1\frac{(x^2+x)\ln(x)}{(1-x^2)(1+x^2)(x^2+x+1)}dx=\frac{-1296G-115\pi^2+192\psi\left(\frac{1}{3}\right)-96\psi\left(\frac{2}{3}\right)}{2592}.$$
My questions are: What is the closed form of this integral, if there is any? And are there any methods or techniques that would be effective at attacking integrals with multiple logarithms in the numerator of the integrand?
Edit: G is Catalan's constant and psi is the digamma function.