\[ y''(x)=-\frac {A y(x)}{\left (a x^2+b x+c\right )^2} \] ✓ Mathematica : cpu = 0.903225 (sec), leaf count = 211
\[\left \{\left \{y(x)\to \frac {c_2 \sqrt {a x^2+b x+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}+c_1 \sqrt {x (a x+b)+c} \exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right \}\right \}\] ✓ Maple : cpu = 0.282 (sec), leaf count = 178
\[ \left \{ y \left ( x \right ) =\sqrt {a{x}^{2}+bx+c} \left ( \left ( { \left ( i\sqrt {4\,ac-{b}^{2}}-2\,ax-b \right ) \left ( 2\,ax+b+i\sqrt {4\,ac-{b}^{2}} \right ) ^{-1}} \right ) ^{{\frac {a}{2}\sqrt {{\frac {-4\,ac+{b}^{2}-4\,A}{{a}^{2}}}}{\frac {1}{\sqrt {-4\,ac+{b}^{2}}}}}}{\it \_C1}+ \left ( { \left ( i\sqrt {4\,ac-{b}^{2}}-2\,ax-b \right ) \left ( 2\,ax+b+i\sqrt {4\,ac-{b}^{2}} \right ) ^{-1}} \right ) ^{-{\frac {a}{2}\sqrt {{\frac {-4\,ac+{b}^{2}-4\,A}{{a}^{2}}}}{\frac {1}{\sqrt {-4\,ac+{b}^{2}}}}}}{\it \_C2} \right ) \right \} \]