ODE No. 1378

$$y^{″}(x)=-\frac{2(x^2-1)y^{\prime }(x)}{(x-1{)}^{2}x}-\frac{(-2x^2+2x+2)y(x)}{(x-1{)}^2{x}^2}$$ ✓ 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]
 

$$\left\{\{y(x)\to \frac{c_1{x}^2}{1-x}+\frac{c_{2}x(2x^2\log (1-x)-2x^2\log (x)+2x-2x\log (1-x)+2x\log (x)-1)}{(x-1{)}^2}\}\right\}$$ ✓ 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))
 

$$y\left(x\right)=\frac{(-c_{2}x(x-1)\ln (x-1)+c_{2}x(x-1)\ln \left(x\right)+c_1{x}^2+(-c_1-c_2)x+\frac{c_2}{2})x}{{(x-1)}^2}$$