\[ y''(x)=-\frac {27 x y(x)}{16 \left (x^3-1\right )^2} \] ✓ Mathematica : cpu = 1.47195 (sec), leaf count = 258
DSolve[Derivative[2][y][x] == (-27*x*y[x])/(16*(-1 + x^3)^2),y[x],x]
\[\left \{\left \{y(x)\to \frac {\sqrt {2} c_2 (1-x)^{3/4} \sqrt [4]{x^2+x+1} \int _1^x\frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}}dK[1]}{\sqrt [4]{\sqrt {3} x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+\sqrt {3}}}+\frac {\sqrt {2} c_1 (1-x)^{3/4} \sqrt [4]{x^2+x+1}}{\sqrt [4]{\sqrt {3} x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+\sqrt {3}}}\right \}\right \}\] ✓ Maple : cpu = 0.137 (sec), leaf count = 44
dsolve(diff(diff(y(x),x),x) = -27/16*x/(x^3-1)^2*y(x),y(x))
\[y \left (x \right ) = \sqrt {x}\, \left (x^{3}-1\right )^{\frac {1}{4}} \left (\LegendreP \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) c_{1}+\LegendreQ \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) c_{2}\right )\]