Closed form for ${\large\int}_0^1\frac{\ln^3x}{\sqrt{x^2-x+1}}dx$

I followed the same approach as I used in an answer to another question, and expanded your integral in multiple polylogarithms of weight 4, then used some patterns in their values of weight 3 to guess terms that might appear in the integral. Then I used an integer relation algorithm to express your integral in terms of logs, zeta functions and polylogarithms of small rational arguments with a tolerance of about $10^{-200}$, and I found this value, which is correct to $3000$ digits.

There are $27$ polylogarithm terms there in total, and while I managed to simplify them somewhat, I never quite managed to evaluate some of them except by an integer relation algorithm.

Here it is: $$\textstyle\def\Li{\mathrm{Li}} -6 \Li_2(\frac{1}{3}) \zeta (2)+27 \Li_4(\frac{3}{4})+36 \Li_4(\frac{2}{3})-4 \Li_4(\frac{1}{2})-18 \Li_4(\frac{1}{3})-\frac{9}{2} \Li_4(\frac{1}{4})+6 \Li_2(\frac{1}{3}){}^2+6 \Li_2(\frac{1}{3}) \log ^2 3+24 \Li_2(\frac{1}{3}) \log ^2 2-48 \Li_3(\frac{2}{3}) \log3+96 \Li_3(\frac{2}{3}) \log2-48 \Li_3(\frac{1}{3}) \log3+72 \Li_3(\frac{1}{3}) \log2-24 \Li_2(\frac{1}{3}) \log2 \log3+78 \zeta (3) \log3-142 \zeta (3) \log2-\frac{151}{4} \zeta (4)-69 \zeta (2) \log ^2 3-122 \zeta (2) \log ^2 2+192 \zeta (2) \log2 \log3+\frac{73}{4} \log ^4 3+\frac{89}{6} \log ^4 2-70 \log2 \log ^3 3-56 \log ^3 2 \log3+93 \log ^2 2 \log ^2 3 $$

Here is the equivalent Mathematica expression to save people typing:

(-151*Pi^4)/360 - (61*Pi^2*Log[2]^2)/3 + (89*Log[2]^4)/6 + 32*Pi^2*Log[2]*Log[3] -  56*Log[2]^3*Log[3] - (23*Pi^2*Log[3]^2)/2 + 93*Log[2]^2*Log[3]^2 - 70*Log[2]*Log[3]^3 +  (73*Log[3]^4)/4 - Pi^2*PolyLog[2, 1/3] + 24*Log[2]^2*PolyLog[2, 1/3] -  24*Log[2]*Log[3]*PolyLog[2, 1/3] + 6*Log[3]^2*PolyLog[2, 1/3] + 6*PolyLog[2, 1/3]^2 +  72*Log[2]*PolyLog[3, 1/3] - 48*Log[3]*PolyLog[3, 1/3] + 96*Log[2]*PolyLog[3, 2/3] -  48*Log[3]*PolyLog[3, 2/3] - (9*PolyLog[4, 1/4])/2 - 18*PolyLog[4, 1/3] - 4*PolyLog[4, 1/2] +  36*PolyLog[4, 2/3] + 27*PolyLog[4, 3/4] - 142*Log[2]*Zeta[3] + 78*Log[3]*Zeta[3]