\[ y''(x)=-\frac {2 \left (x^2-1\right ) y'(x)}{(x-1)^2 x}-\frac {\left (-2 x^2+2 x+2\right ) y(x)}{(x-1)^2 x^2} \] ✓ Mathematica : cpu = 0.0573155 (sec), leaf count = 56
\[\left \{\left \{y(x)\to -\frac {x \left (c_1 x^2-c_1 x-2 c_2 x-2 c_2 (x-1) x \log (1-x)+2 c_2 (x-1) x \log (x)+c_2\right )}{(x-1)^2}\right \}\right \}\]
✓ Maple : cpu = 0.06 (sec), leaf count = 48
\[ \left \{ y \left ( x \right ) ={\frac {x}{ \left ( x-1 \right ) ^{2}} \left ( -{\it \_C2}\,x \left ( x-1 \right ) \ln \left ( x-1 \right ) +{\it \_C2}\,x \left ( x-1 \right ) \ln \left ( x \right ) +{\it \_C1}\,{x}^{2}+ \left ( -{\it \_C1}-{\it \_C2} \right ) x+{\frac {{\it \_C2}}{2}} \right ) } \right \} \]