\[ y''(x) \left (a x^2+b x+c\right )+(d x+f) y'(x)+g y(x)=0 \] ✓ Mathematica : cpu = 5.12741 (sec), leaf count = 498
\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (-\frac {a-d+\sqrt {(a-d)^2-4 a g}}{2 a},\frac {-a+d+\sqrt {(a-d)^2-4 a g}}{2 a};\frac {\left (b+\sqrt {b^2-4 a c}\right ) d-2 a f}{2 a \sqrt {b^2-4 a c}};\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )-c_2 2^{\frac {\frac {b d}{\sqrt {b^2-4 a c}}+d}{2 a}-\frac {f}{\sqrt {b^2-4 a c}}-1} \exp \left (-\frac {i \pi \left (d \left (\sqrt {b^2-4 a c}+b\right )-2 a f\right )}{2 a \sqrt {b^2-4 a c}}\right ) \left (\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}}\right )^{-\frac {\frac {b d}{\sqrt {b^2-4 a c}}+d}{2 a}+\frac {f}{\sqrt {b^2-4 a c}}+1} \, _2F_1\left (\frac {\frac {2 f a}{\sqrt {b^2-4 a c}}+a-\frac {b d}{\sqrt {b^2-4 a c}}-\sqrt {(a-d)^2-4 a g}}{2 a},\frac {\frac {2 f a}{\sqrt {b^2-4 a c}}+a-\frac {b d}{\sqrt {b^2-4 a c}}+\sqrt {(a-d)^2-4 a g}}{2 a};-\frac {\frac {b d}{\sqrt {b^2-4 a c}}+d+a \left (-\frac {2 f}{\sqrt {b^2-4 a c}}-4\right )}{2 a};\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )\right \}\right \}\] ✓ Maple : cpu = 0.215 (sec), leaf count = 501
\[\left \{y \left (x \right ) = c_{2} \left (2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2} x +\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a b -4 a c +b^{2}\right )^{\frac {a f -\frac {b d}{2}+\left (a -\frac {d}{2}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a}{\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}} \hypergeom \left (\left [\frac {2 a f -b d +\left (a +\sqrt {a^{2}+d^{2}+\left (-2 d -4 g \right ) a}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a}{2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}, \frac {2 a f -b d +\left (a -\sqrt {a^{2}+d^{2}+\left (-2 d -4 g \right ) a}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a}{2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}\right ], \left [\frac {4 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}-\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a d +2 a f -b d}{2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}\right ], \frac {4 a c -b^{2}+\left (-2 a^{2} x -a b \right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}{8 a c -2 b^{2}}\right )+c_{1} \hypergeom \left (\left [\frac {-a +d +\sqrt {a^{2}+d^{2}+\left (-2 d -4 g \right ) a}}{2 a}, -\frac {a -d +\sqrt {a^{2}+d^{2}+\left (-2 d -4 g \right ) a}}{2 a}\right ], \left [\frac {\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a d -2 a f +b d}{2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}\right ], \frac {4 a c -b^{2}+\left (-2 a^{2} x -a b \right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}{8 a c -2 b^{2}}\right )\right \}\]