\[ y''(x)=-\frac {27 x y(x)}{16 \left (x^3-1\right )^2} \] ✓ Mathematica : cpu = 31.941 (sec), leaf count = 2622
DSolve[Derivative[2][y][x] == (-27*x*y[x])/(16*(-1 + x^3)^2),y[x],x]
\[\left \{\left \{y(x)\to -\frac {e^{-\frac {1}{2} \text {arctanh}\left (\frac {-2 x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+2}{2 \sqrt {3}}\right )} (x-1) \left (-2 \left (-i+\sqrt {3}\right ) x-i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+\sqrt {2 x+i \sqrt {3}+1} \sqrt {6 x-3 i \sqrt {3}+3}+4 i\right )^{\frac {5}{32} \left (1-i \sqrt {3}\right )} \left (-2 \left (9 i+5 \sqrt {3}\right ) x+9 i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+5 \sqrt {2 x+i \sqrt {3}+1} \sqrt {6 x-3 i \sqrt {3}+3}+4 \sqrt {3}-24 i\right )^{\frac {1}{32} \left (1+i \sqrt {3}\right )} \left (\left (256985009468147850 i-163072339831481354 \sqrt {3}\right ) x-128492504734073925 i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+81536169915740677 \sqrt {2 x+i \sqrt {3}+1} \sqrt {6 x-3 i \sqrt {3}+3}-210028674649814602 \sqrt {3}-116116005013148106 i\right )^{\frac {1}{32} \left (15-i \sqrt {3}\right )} c_1 \left (\left (3974 i-4610 \sqrt {3}\right ) x+\left (-1987 i+2305 \sqrt {3}\right ) \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-2 \sqrt {2 \left (-9867779-707709 i \sqrt {3}\right )}\right )^{\frac {1}{32} \left (11+5 i \sqrt {3}\right )}}{\sqrt {2} \sqrt {(x-1) \left (-2 x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-1\right )}}-\frac {e^{-\frac {1}{2} \text {arctanh}\left (\frac {-2 x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+2}{2 \sqrt {3}}\right )} (x-1) \left (-2 \sqrt {3} x+2 i x+\sqrt {3} \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+4 i\right )^{5/32} \left (-2 \left (-i+\sqrt {3}\right ) x+\sqrt {3} \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+4 i\right )^{-\frac {5 i \sqrt {3}}{32}} \sqrt [32]{-10 \sqrt {3} x-18 i x+5 \sqrt {3} \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+9 i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+4 \sqrt {3}-24 i} \left (-2 \left (9 i+5 \sqrt {3}\right ) x+5 \sqrt {3} \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+9 i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+4 \sqrt {3}-24 i\right )^{\frac {i \sqrt {3}}{32}} \left (-4610 \sqrt {3} x+3974 i x+2305 \sqrt {3} \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-1987 i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-2 \sqrt {2 \left (-9867779-707709 i \sqrt {3}\right )}\right )^{11/32} \left (-163072339831481354 \sqrt {3} x+256985009468147850 i x+81536169915740677 \sqrt {3} \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-128492504734073925 i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-210028674649814602 \sqrt {3}-116116005013148106 i\right )^{15/32} \left (\left (256985009468147850 i-163072339831481354 \sqrt {3}\right ) x+81536169915740677 \sqrt {3} \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-128492504734073925 i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-210028674649814602 \sqrt {3}-116116005013148106 i\right )^{-\frac {i \sqrt {3}}{32}} c_2 \int _1^x\frac {2 e^{\text {arctanh}\left (\frac {-2 K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+2}{2 \sqrt {3}}\right )} \left (-2 K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-1\right ) \left (-2 \left (-i+\sqrt {3}\right ) K[1]+\sqrt {3} \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-i \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+4 i\right )^{\frac {5 i \sqrt {3}}{16}} \left (-2 \left (9 i+5 \sqrt {3}\right ) K[1]+5 \sqrt {3} \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+9 i \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+4 \sqrt {3}-24 i\right )^{-\frac {i \sqrt {3}}{16}} \left (\left (256985009468147850 i-163072339831481354 \sqrt {3}\right ) K[1]+81536169915740677 \sqrt {3} \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-128492504734073925 i \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-210028674649814602 \sqrt {3}-116116005013148106 i\right )^{\frac {i \sqrt {3}}{16}} \left (\left (3974 i-4610 \sqrt {3}\right ) K[1]+\left (-1987 i+2305 \sqrt {3}\right ) \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-2 \sqrt {2 \left (-9867779-707709 i \sqrt {3}\right )}\right )^{-\frac {5 i \sqrt {3}}{16}}}{(K[1]-1) \left (-2 \sqrt {3} K[1]+2 i K[1]+\sqrt {3} \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-i \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+4 i\right )^{5/16} \sqrt [16]{-10 \sqrt {3} K[1]-18 i K[1]+5 \sqrt {3} \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+9 i \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+4 \sqrt {3}-24 i} \left (-4610 \sqrt {3} K[1]+3974 i K[1]+2305 \sqrt {3} \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-1987 i \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-2 \sqrt {2 \left (-9867779-707709 i \sqrt {3}\right )}\right )^{11/16} \left (-163072339831481354 \sqrt {3} K[1]+256985009468147850 i K[1]+81536169915740677 \sqrt {3} \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-128492504734073925 i \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-210028674649814602 \sqrt {3}-116116005013148106 i\right )^{15/16}}dK[1] \left (\left (3974 i-4610 \sqrt {3}\right ) x+\left (-1987 i+2305 \sqrt {3}\right ) \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-2 \sqrt {2 \left (-9867779-707709 i \sqrt {3}\right )}\right )^{\frac {5 i \sqrt {3}}{32}}}{\sqrt {2} \sqrt {(x-1) \left (-2 x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-1\right )}}\right \}\right \}\] ✓ Maple : cpu = 0.095 (sec), leaf count = 44
dsolve(diff(diff(y(x),x),x) = -27/16*x/(x^3-1)^2*y(x),y(x))
\[y \left (x \right ) = \sqrt {x}\, \left (x^{3}-1\right )^{\frac {1}{4}} \left (\operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) c_{1}+\operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) c_{2}\right )\]