\[ \boxed { \left ( a{x}^{2}+bx+c \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( dx+f \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +gy \left ( x \right ) =0} \]
Mathematica: cpu = 14.510843 (sec), leaf count = 56 \[ \left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{g \unicode {f818}(\unicode {f817})+(\unicode {f817} d+f) \unicode {f818}'(\unicode {f817})+\left (a \unicode {f817}^2+b \unicode {f817}+c\right ) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \} \]
Maple: cpu = 0.140 (sec), leaf count = 501 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, {\mbox {$_2$F$_1$}({\frac {1}{2\,a} \left ( -a+d+\sqrt {{a}^{2}+ \left ( -2\,d-4\,g \right ) a+{d}^{2}} \right ) },-{\frac {1}{2\,a} \left ( a-d+\sqrt {{a}^{2}+ \left ( -2\,d-4\,g \right ) a+{d}^{2}} \right ) };\,{\frac {1}{2\,{a}^{2}} \left ( d\sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}a-2\,af+bd \right ) {\frac {1}{\sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}}}};\,{\frac {1}{8\,ac-2\,{b}^{2}} \left ( \left ( -2\,{a}^{2}x-ab \right ) \sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}+4\,ac-{b}^{2} \right ) })} +{\it \_C2}\, \left ( 2\,\sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}x{a}^ {2}+\sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}ba-4\,ac+{b}^{2} \right ) ^{{\frac {1}{{a}^{2}} \left ( a \left ( a-{\frac {d}{2}} \right ) \sqrt { {\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}+af-{\frac {bd}{2}} \right ) {\frac { 1}{\sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}}}}} {\mbox {$_2$F$_1$}({\frac {1}{2\,{a}^{2}} \left ( a \left ( a+\sqrt {{a}^{2}+ \left ( -2\,d-4\,g \right ) a+{d}^{2}} \right ) \sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}+2\,af-bd \right ) {\frac {1}{\sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}}}},{\frac {1}{2\,{a}^{2}} \left ( a \left ( a-\sqrt {{a}^{2}+ \left ( -2\,d-4\,g \right ) a+{d}^{2}} \right ) \sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}+2\,af-bd \right ) {\frac {1}{\sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}}}};\,{\frac {1}{2\,{a}^{2}} \left ( 4\,{a}^{2}\sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}-d\sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}a+2\,af-bd \right ) {\frac {1}{\sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}}}};\,{\frac {1}{8\,ac-2\,{b}^{2}} \left ( \left ( -2\,{a}^{2}x-ab \right ) \sqrt {{\frac {-4\,ac+{b}^{2}}{{a}^{2}}}}+4\,ac-{b}^{2} \right ) })} \right \} \]