\[ \left (a x^2+1\right ) y''(x)+b x y'(x)+c y(x)=0 \] ✓ Mathematica : cpu = 0.0514049 (sec), leaf count = 162
DSolve[c*y[x] + b*x*Derivative[1][y][x] + (1 + a*x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (a x^2+1\right )^{\frac {2 a-b}{4 a}} P_{\frac {\sqrt {a^2-2 b a-4 c a+b^2}-a}{2 a}}^{\frac {b-2 a}{2 a}}\left (i \sqrt {a} x\right )+c_2 \left (a x^2+1\right )^{\frac {2 a-b}{4 a}} Q_{\frac {\sqrt {a^2-2 b a-4 c a+b^2}-a}{2 a}}^{\frac {b-2 a}{2 a}}\left (i \sqrt {a} x\right )\right \}\right \}\] ✓ Maple : cpu = 0.11 (sec), leaf count = 124
dsolve((a*x^2+1)*diff(diff(y(x),x),x)+b*x*diff(y(x),x)+c*y(x)=0,y(x))
\[y \left (x \right ) = \left (a \,x^{2}+1\right )^{\frac {2 a -b}{4 a}} \left (\LegendreP \left (\frac {\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-a}{2 a}, \frac {2 a -b}{2 a}, \sqrt {-a}\, x \right ) c_{1}+\LegendreQ \left (\frac {\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-a}{2 a}, \frac {2 a -b}{2 a}, \sqrt {-a}\, x \right ) c_{2}\right )\]