\[ a y(x)+\left (x^2+1\right ) y''(x)+3 x y'(x)=0 \] ✓ Mathematica : cpu = 0.0110828 (sec), leaf count = 82
DSolve[a*y[x] + 3*x*Derivative[1][y][x] + (1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {c_1 P_{\frac {1}{2} \left (2 \sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)}{\sqrt [4]{x^2+1}}+\frac {c_2 Q_{\frac {1}{2} \left (2 \sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)}{\sqrt [4]{x^2+1}}\right \}\right \}\] ✓ Maple : cpu = 0.093 (sec), leaf count = 53
dsolve((x^2+1)*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+a*y(x)=0,y(x))
\[y \left (x \right ) = \frac {c_{2} \left (x +\sqrt {x^{2}+1}\right )^{-\sqrt {1-a}}+c_{1} \left (x +\sqrt {x^{2}+1}\right )^{\sqrt {1-a}}}{\sqrt {x^{2}+1}}\]