\[ a x y'(x)+(a-2) y(x)+\left (x^2+1\right ) y''(x)=0 \] ✓ Mathematica : cpu = 0.0183044 (sec), leaf count = 82
DSolve[(-2 + a)*y[x] + a*x*Derivative[1][y][x] + (1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (x^2+1\right )^{\frac {2-a}{4}} P_{\frac {a-4}{2}}^{\frac {a-2}{2}}(i x)+c_2 \left (x^2+1\right )^{\frac {2-a}{4}} Q_{\frac {a-4}{2}}^{\frac {a-2}{2}}(i x)\right \}\right \}\] ✓ Maple : cpu = 0.102 (sec), leaf count = 36
dsolve((x^2+1)*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(a-2)*y(x)=0,y(x))
\[y \left (x \right ) = c_{1} \left (x^{2}+1\right )^{1-\frac {a}{2}}+c_{2} \hypergeom \left (\left [1, \frac {a}{2}-\frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) x\]