2.184   ODE No. 184

$\left (y'(x)+y(x)^2\right ) \left (a x^2+b x+c\right )^2+A=0$ Mathematica : cpu = 1.1501 (sec), leaf count = 704

$\left \{\left \{y(x)\to -\frac {c_1 \left (\frac {2 a \sqrt {1-\frac {4 A}{b^2-4 a c}} \sqrt {x (a x+b)+c} \exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (\frac {(2 a x+b)^2}{4 a c-b^2}+1\right )}+\frac {(2 a x+b) \exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {x (a x+b)+c}}\right )-\frac {2 a \sqrt {a x^2+b x+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (\frac {(2 a x+b)^2}{4 a c-b^2}+1\right )}+\frac {(2 a x+b) \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \sqrt {a x^2+b x+c}}}{c_1 \sqrt {x (a x+b)+c} \left (-\exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right )-\frac {\sqrt {a x^2+b x+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}}\right \}\right \}$ Maple : cpu = 0.302 (sec), leaf count = 493

$\left \{ y \left ( x \right ) =2\,{\frac {a}{\sqrt {-4\,ac+{b}^{2}} \left ( 2\,ax+b+i\sqrt {4\,ac-{b}^{2}} \right ) \left ( i\sqrt {4\,ac-{b}^{2}}-2\,ax-b \right ) } \left ( \left ( i\sqrt {{\frac {-4\,ac+{b}^{2}-4\,A}{{a}^{2}}}}a\sqrt {4\,ac-{b}^{2}}-2\, \left ( ax+b/2 \right ) \sqrt {-4\,ac+{b}^{2}} \right ) {\it \_C1}\, \left ( {\frac {i\sqrt {4\,ac-{b}^{2}}-2\,ax-b}{2\,ax+b+i\sqrt {4\,ac-{b}^{2}}}} \right ) ^{-1/2\,{\frac {a}{\sqrt {-4\,ac+{b}^{2}}}\sqrt {{\frac {-4\,ac+{b}^{2}-4\,A}{{a}^{2}}}}}}- \left ( i\sqrt {{\frac {-4\,ac+{b}^{2}-4\,A}{{a}^{2}}}}a\sqrt {4\,ac-{b}^{2}}+2\, \left ( ax+b/2 \right ) \sqrt {-4\,ac+{b}^{2}} \right ) \left ( {\frac {i\sqrt {4\,ac-{b}^{2}}-2\,ax-b}{2\,ax+b+i\sqrt {4\,ac-{b}^{2}}}} \right ) ^{1/2\,{\frac {a}{\sqrt {-4\,ac+{b}^{2}}}\sqrt {{\frac {-4\,ac+{b}^{2}-4\,A}{{a}^{2}}}}}} \right ) \left ( {\it \_C1}\, \left ( {\frac {i\sqrt {4\,ac-{b}^{2}}-2\,ax-b}{2\,ax+b+i\sqrt {4\,ac-{b}^{2}}}} \right ) ^{-1/2\,{\frac {a}{\sqrt {-4\,ac+{b}^{2}}}\sqrt {{\frac {-4\,ac+{b}^{2}-4\,A}{{a}^{2}}}}}}+ \left ( {\frac {i\sqrt {4\,ac-{b}^{2}}-2\,ax-b}{2\,ax+b+i\sqrt {4\,ac-{b}^{2}}}} \right ) ^{1/2\,{\frac {a}{\sqrt {-4\,ac+{b}^{2}}}\sqrt {{\frac {-4\,ac+{b}^{2}-4\,A}{{a}^{2}}}}}} \right ) ^{-1}} \right \}$