\[ 2 a x y'(x)+(a-1) a y(x)+\left (x^2-1\right ) y''(x)=0 \] ✓ Mathematica : cpu = 0.200208 (sec), leaf count = 132
DSolve[(-1 + a)*a*y[x] + 2*a*x*Derivative[1][y][x] + (-1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {c_2 (x+1)^{\frac {1}{2} \sqrt {(a-1)^2}+\frac {1}{2}} \left (x^2-1\right )^{-a/2} (1-x)^{\frac {1}{2}-\frac {1}{2} \sqrt {(a-1)^2}}}{2 \sqrt {(a-1)^2}}+c_1 (x+1)^{\frac {1}{2}-\frac {1}{2} \sqrt {(a-1)^2}} \left (x^2-1\right )^{-a/2} (1-x)^{\frac {1}{2} \left (\sqrt {(a-1)^2}+1\right )}\right \}\right \}\] ✓ Maple : cpu = 0.031 (sec), leaf count = 27
dsolve((x^2-1)*diff(diff(y(x),x),x)+2*a*x*diff(y(x),x)+a*(a-1)*y(x)=0,y(x))
\[y \left (x \right ) = c_{1} \left (1+x \right )^{1-a}+c_{2} \left (x -1\right )^{1-a}\]