\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-{\frac { \left ( a{x}^{2}+a-1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{x \left ( {x}^{2}+1 \right ) }}-{\frac { \left ( b{x}^{2}+c \right ) y \left ( x \right ) }{{x}^{2} \left ( {x}^{2}+1 \right ) }}=0} \]
Mathematica: cpu = 0.665585 (sec), leaf count = 288 \[ \left \{\left \{y(x)\to c_1 x^{\frac {1}{2} \left (-\sqrt {a^2-4 a-4 c+4}-a+2\right )} \, _2F_1\left (-\frac {1}{4} \sqrt {a^2-2 a-4 b+1}-\frac {1}{4} \sqrt {a^2-4 a-4 c+4}+\frac {1}{4},\frac {1}{4} \sqrt {a^2-2 a-4 b+1}-\frac {1}{4} \sqrt {a^2-4 a-4 c+4}+\frac {1}{4};1-\frac {1}{2} \sqrt {a^2-4 a-4 c+4};-x^2\right )+c_2 x^{\frac {1}{2} \left (\sqrt {a^2-4 a-4 c+4}-a+2\right )} \, _2F_1\left (-\frac {1}{4} \sqrt {a^2-2 a-4 b+1}+\frac {1}{4} \sqrt {a^2-4 a-4 c+4}+\frac {1}{4},\frac {1}{4} \sqrt {a^2-2 a-4 b+1}+\frac {1}{4} \sqrt {a^2-4 a-4 c+4}+\frac {1}{4};\frac {1}{2} \sqrt {a^2-4 a-4 c+4}+1;-x^2\right )\right \}\right \} \]
Maple: cpu = 0.094 (sec), leaf count = 103 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{1-{\frac {a}{2}}}{\it LegendreP} \left ( -{\frac {1}{2}}+{\frac {1}{2}\sqrt {{a}^{2}-2\,a-4\, b+1}},{\frac {1}{2}\sqrt {{a}^{2}-4\,a-4\,c+4}},\sqrt {{x}^{2}+1} \right ) +{\it \_C2}\,{x}^{1-{\frac {a}{2}}}{\it LegendreQ} \left ( -{ \frac {1}{2}}+{\frac {1}{2}\sqrt {{a}^{2}-2\,a-4\,b+1}},{\frac {1}{2} \sqrt {{a}^{2}-4\,a-4\,c+4}},\sqrt {{x}^{2}+1} \right ) \right \} \]