\[ 8 \left (1-x^3\right ) y(x) y''(x)-4 \left (1-x^3\right ) y'(x)^2-12 x^2 y(x) y'(x)+3 x y(x)^2=0 \] ✓ Mathematica : cpu = 6.63326 (sec), leaf count = 1743
\[\left \{\left \{y(x)\to e^{\int _1^x-\frac {2 \left (-\frac {3 (1-K[2])^{3/4} \sqrt [4]{K[2]^2+K[2]+1} \int _1^{K[2]}\frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}}dK[1] K[2]^2}{2 \sqrt {2} \left (K[2]^3-1\right )^{5/4} \sqrt [4]{\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}}}+c_1 \left (-\frac {3 (1-K[2])^{3/4} \sqrt [4]{K[2]^2+K[2]+1} K[2]^2}{2 \sqrt {2} \left (K[2]^3-1\right )^{5/4} \sqrt [4]{\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}}}-\frac {3 \sqrt [4]{K[2]^2+K[2]+1}}{2 \sqrt {2} \sqrt [4]{1-K[2]} \sqrt [4]{K[2]^3-1} \sqrt [4]{\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}}}+\frac {(1-K[2])^{3/4} (2 K[2]+1)}{2 \sqrt {2} \left (K[2]^2+K[2]+1\right )^{3/4} \sqrt [4]{K[2]^3-1} \sqrt [4]{\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}}}-\frac {(1-K[2])^{3/4} \sqrt [4]{K[2]^2+K[2]+1} \left (\frac {\sqrt {2 K[2]-i \sqrt {3}+1}}{\sqrt {2 K[2]+i \sqrt {3}+1}}+\sqrt {3}+\frac {\sqrt {2 K[2]+i \sqrt {3}+1}}{\sqrt {2 K[2]-i \sqrt {3}+1}}\right )}{2 \sqrt {2} \sqrt [4]{K[2]^3-1} \left (\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}\right )^{5/4}}\right )-\frac {3 \sqrt [4]{K[2]^2+K[2]+1} \int _1^{K[2]}\frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}}dK[1]}{2 \sqrt {2} \sqrt [4]{1-K[2]} \sqrt [4]{K[2]^3-1} \sqrt [4]{\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}}}+\frac {(1-K[2])^{3/4} (2 K[2]+1) \int _1^{K[2]}\frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}}dK[1]}{2 \sqrt {2} \left (K[2]^2+K[2]+1\right )^{3/4} \sqrt [4]{K[2]^3-1} \sqrt [4]{\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}}}-\frac {(1-K[2])^{3/4} \sqrt [4]{K[2]^2+K[2]+1} \left (\frac {\sqrt {2 K[2]-i \sqrt {3}+1}}{\sqrt {2 K[2]+i \sqrt {3}+1}}+\sqrt {3}+\frac {\sqrt {2 K[2]+i \sqrt {3}+1}}{\sqrt {2 K[2]-i \sqrt {3}+1}}\right ) \int _1^{K[2]}\frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}}dK[1]}{2 \sqrt {2} \sqrt [4]{K[2]^3-1} \left (\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}\right )^{5/4}}+\frac {\sqrt [4]{\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}}}{\sqrt {2} (1-K[2])^{3/4} \sqrt [4]{K[2]^2+K[2]+1} \sqrt [4]{K[2]^3-1}}\right )}{-\frac {\sqrt {2} (1-K[2])^{3/4} \sqrt [4]{K[2]^2+K[2]+1} c_1}{\sqrt [4]{K[2]^3-1} \sqrt [4]{\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}}}-\frac {\sqrt {2} (1-K[2])^{3/4} \sqrt [4]{K[2]^2+K[2]+1} \int _1^{K[2]}\frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}}dK[1]}{\sqrt [4]{K[2]^3-1} \sqrt [4]{\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}}}}dK[2]} c_2\right \}\right \}\] ✓ Maple : cpu = 1.175 (sec), leaf count = 49
\[y \relax (x ) = \frac {\left (c_{1} \LegendreQ \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (x -1\right ) \left (x^{2}+x +1\right )}\right )+\frac {c_{2} \LegendreP \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (x -1\right ) \left (x^{2}+x +1\right )}\right )}{2}\right )^{2} x}{c_{1}}\]