\[ \left (a x^2+1\right ) y''(x)+b x y'(x)+c y(x)=0 \] ✓ Mathematica : cpu = 0.0553403 (sec), leaf count = 162
\[\left \{\left \{y(x)\to c_1 \left (a x^2+1\right )^{\frac {2 a-b}{4 a}} P_{\frac {\sqrt {a^2-2 b a-4 c a+b^2}-a}{2 a}}^{\frac {b-2 a}{2 a}}\left (i \sqrt {a} x\right )+c_2 \left (a x^2+1\right )^{\frac {2 a-b}{4 a}} Q_{\frac {\sqrt {a^2-2 b a-4 c a+b^2}-a}{2 a}}^{\frac {b-2 a}{2 a}}\left (i \sqrt {a} x\right )\right \}\right \}\] ✓ Maple : cpu = 0.444 (sec), leaf count = 124
\[ \left \{ y \left ( x \right ) = \left ( a{x}^{2}+1 \right ) ^{{\frac {2\,a-b}{4\,a}}} \left ( {\it LegendreP} \left ( {\frac {1}{2\,a} \left ( \sqrt {{a}^{2}+ \left ( -2\,b-4\,c \right ) a+{b}^{2}}-a \right ) },{\frac {2\,a-b}{2\,a}},\sqrt {-a}x \right ) {\it \_C1}+{\it LegendreQ} \left ( {\frac {1}{2\,a} \left ( \sqrt {{a}^{2}+ \left ( -2\,b-4\,c \right ) a+{b}^{2}}-a \right ) },{\frac {2\,a-b}{2\,a}},\sqrt {-a}x \right ) {\it \_C2} \right ) \right \} \]