\[ y''(x)=-\frac {2 \left (x^2-1\right ) y'(x)}{(x-1)^2 x}-\frac {\left (-2 x^2+2 x+2\right ) y(x)}{(x-1)^2 x^2} \] ✓ Mathematica : cpu = 0.0366335 (sec), leaf count = 65
DSolve[Derivative[2][y][x] == -(((2 + 2*x - 2*x^2)*y[x])/((-1 + x)^2*x^2)) - (2*(-1 + x^2)*Derivative[1][y][x])/((-1 + x)^2*x),y[x],x]
\[\left \{\left \{y(x)\to \frac {c_1 x^2}{1-x}+\frac {c_2 x \left (2 x^2 \log (1-x)-2 x^2 \log (x)+2 x-2 x \log (1-x)+2 x \log (x)-1\right )}{(x-1)^2}\right \}\right \}\] ✓ Maple : cpu = 0.034 (sec), leaf count = 48
dsolve(diff(diff(y(x),x),x) = -2/x*(x^2-1)/(x-1)^2*diff(y(x),x)-(-2*x^2+2*x+2)/x^2/(x-1)^2*y(x),y(x))
\[y \left (x \right ) = \frac {\left (-c_{2} x \left (x -1\right ) \ln \left (x -1\right )+c_{2} x \left (x -1\right ) \ln \left (x \right )+c_{1} x^{2}+\left (-c_{1}-c_{2}\right ) x +\frac {c_{2}}{2}\right ) x}{\left (x -1\right )^{2}}\]