\[ \boxed { \left ( a{x}^{2}+bx+c \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2} \right ) +A=0} \]
Mathematica: cpu = 1.467686 (sec), leaf count = 704 \[ \left \{\left \{y(x)\to -\frac {-\frac {2 a \sqrt {a x^2+b x+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (\frac {(2 a x+b)^2}{4 a c-b^2}+1\right )}+\frac {(2 a x+b) \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \sqrt {a x^2+b x+c}}+c_1 \left (\frac {2 a \sqrt {1-\frac {4 A}{b^2-4 a c}} \sqrt {x (a x+b)+c} \exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (\frac {(2 a x+b)^2}{4 a c-b^2}+1\right )}+\frac {(2 a x+b) \exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {x (a x+b)+c}}\right )}{c_1 \sqrt {x (a x+b)+c} \left (-\exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right )-\frac {\sqrt {a x^2+b x+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}}\right \}\right \} \]
Maple: cpu = 0.234 (sec), leaf count = 846 \[ \left \{ y \left ( x \right ) =-2\,{\frac {a}{\sqrt {-4\,ac+{b}^{2}} \left ( 2\,ax+b+i\sqrt {4\,ac-{b}^{2}} \right ) \left ( i\sqrt {4\,ac-{ b}^{2}}-2\,ax-b \right ) } \left ( -i\sqrt {4\,ac-{b}^{2}} \left ( { \frac {i\sqrt {4\,ac-{b}^{2}}-2\,ax-b}{2\,ax+b+i\sqrt {4\,ac-{b}^{2}}} } \right ) ^{-1/2\,{\frac {a}{\sqrt {-4\,ac+{b}^{2}}}\sqrt {-{\frac {4 \,ac-{b}^{2}+4\,A}{{a}^{2}}}}}}\sqrt {-{\frac {4\,ac-{b}^{2}+4\,A}{{a} ^{2}}}}{\it \_C1}\,a+i\sqrt {4\,ac-{b}^{2}} \left ( {\frac {i\sqrt {4\, ac-{b}^{2}}-2\,ax-b}{2\,ax+b+i\sqrt {4\,ac-{b}^{2}}}} \right ) ^{1/2\,{ \frac {a}{\sqrt {-4\,ac+{b}^{2}}}\sqrt {-{\frac {4\,ac-{b}^{2}+4\,A}{{ a}^{2}}}}}}\sqrt {-{\frac {4\,ac-{b}^{2}+4\,A}{{a}^{2}}}}a+2\,\sqrt {- 4\,ac+{b}^{2}} \left ( {\frac {i\sqrt {4\,ac-{b}^{2}}-2\,ax-b}{2\,ax+b+ i\sqrt {4\,ac-{b}^{2}}}} \right ) ^{-1/2\,{\frac {a}{\sqrt {-4\,ac+{b}^ {2}}}\sqrt {-{\frac {4\,ac-{b}^{2}+4\,A}{{a}^{2}}}}}}{\it \_C1}\,ax+2 \,\sqrt {-4\,ac+{b}^{2}} \left ( {\frac {i\sqrt {4\,ac-{b}^{2}}-2\,ax-b }{2\,ax+b+i\sqrt {4\,ac-{b}^{2}}}} \right ) ^{1/2\,{\frac {a}{\sqrt {-4 \,ac+{b}^{2}}}\sqrt {-{\frac {4\,ac-{b}^{2}+4\,A}{{a}^{2}}}}}}ax+ \sqrt {-4\,ac+{b}^{2}} \left ( {\frac {i\sqrt {4\,ac-{b}^{2}}-2\,ax-b}{ 2\,ax+b+i\sqrt {4\,ac-{b}^{2}}}} \right ) ^{-1/2\,{\frac {a}{\sqrt {-4 \,ac+{b}^{2}}}\sqrt {-{\frac {4\,ac-{b}^{2}+4\,A}{{a}^{2}}}}}}{\it \_C1}\,b+\sqrt {-4\,ac+{b}^{2}} \left ( {\frac {i\sqrt {4\,ac-{b}^{2}}- 2\,ax-b}{2\,ax+b+i\sqrt {4\,ac-{b}^{2}}}} \right ) ^{1/2\,{\frac {a}{ \sqrt {-4\,ac+{b}^{2}}}\sqrt {-{\frac {4\,ac-{b}^{2}+4\,A}{{a}^{2}}}}} }b \right ) \left ( {\it \_C1}\, \left ( {\frac {i\sqrt {4\,ac-{b}^{2}}- 2\,ax-b}{2\,ax+b+i\sqrt {4\,ac-{b}^{2}}}} \right ) ^{-1/2\,{\frac {a}{ \sqrt {-4\,ac+{b}^{2}}}\sqrt {-{\frac {4\,ac-{b}^{2}+4\,A}{{a}^{2}}}}} }+ \left ( {\frac {i\sqrt {4\,ac-{b}^{2}}-2\,ax-b}{2\,ax+b+i\sqrt {4\,a c-{b}^{2}}}} \right ) ^{1/2\,{\frac {a}{\sqrt {-4\,ac+{b}^{2}}}\sqrt {- {\frac {4\,ac-{b}^{2}+4\,A}{{a}^{2}}}}}} \right ) ^{-1}} \right \} \]