##### 4.8.30 $$\left (y'(x)+y(x)^2\right ) \left (a+b x+c x^2\right )^2+A=0$$

ODE
$\left (y'(x)+y(x)^2\right ) \left (a+b x+c x^2\right )^2+A=0$ ODE Classiﬁcation

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 1.3893 (sec), leaf count = 612

$\left \{\left \{y(x)\to \frac {b^2 c_1 \left (-\exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right )+b c_1 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+4 A c_1 \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+4 a c c_1 \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+2 c c_1 x \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+b+2 c x}{2 (a+x (b+c x)) \left (1+c_1 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right )}\right \}\right \}$

Maple
cpu = 0.383 (sec), leaf count = 846

$\left [y \left (x \right ) = -\frac {2 \left (-i \left (\frac {i \sqrt {4 c a -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 c a -b^{2}}}\right )^{-\frac {c \sqrt {-\frac {4 c a -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} \sqrt {-\frac {4 c a -b^{2}+4 A}{c^{2}}}\, \sqrt {4 c a -b^{2}}\, \textit {\_C1} c +i \left (\frac {i \sqrt {4 c a -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 c a -b^{2}}}\right )^{\frac {c \sqrt {-\frac {4 c a -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} \sqrt {-\frac {4 c a -b^{2}+4 A}{c^{2}}}\, \sqrt {4 c a -b^{2}}\, c +2 \left (\frac {i \sqrt {4 c a -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 c a -b^{2}}}\right )^{-\frac {c \sqrt {-\frac {4 c a -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} \sqrt {-4 c a +b^{2}}\, \textit {\_C1} c x +2 \left (\frac {i \sqrt {4 c a -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 c a -b^{2}}}\right )^{\frac {c \sqrt {-\frac {4 c a -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} \sqrt {-4 c a +b^{2}}\, c x +\left (\frac {i \sqrt {4 c a -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 c a -b^{2}}}\right )^{-\frac {c \sqrt {-\frac {4 c a -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} \sqrt {-4 c a +b^{2}}\, \textit {\_C1} b +\left (\frac {i \sqrt {4 c a -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 c a -b^{2}}}\right )^{\frac {c \sqrt {-\frac {4 c a -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} \sqrt {-4 c a +b^{2}}\, b \right ) c}{\sqrt {-4 c a +b^{2}}\, \left (2 c x +b +i \sqrt {4 c a -b^{2}}\right ) \left (i \sqrt {4 c a -b^{2}}-2 c x -b \right ) \left (\textit {\_C1} \left (\frac {i \sqrt {4 c a -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 c a -b^{2}}}\right )^{-\frac {c \sqrt {-\frac {4 c a -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 c a +b^{2}}}}+\left (\frac {i \sqrt {4 c a -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 c a -b^{2}}}\right )^{\frac {c \sqrt {-\frac {4 c a -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 c a +b^{2}}}}\right )}\right ]$ Mathematica raw input

DSolve[A + (a + b*x + c*x^2)^2*(y[x]^2 + y'[x]) == 0,y[x],x]

Mathematica raw output

{{y[x] -> (b + Sqrt[b^2 - 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)] + 2*c*x + 4*A*E^(
(2*Sqrt[-b^2 + 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*ArcTan[(b + 2*c*x)/Sqrt[-b^2
 + 4*a*c]])/Sqrt[b^2 - 4*a*c])*C[1] - b^2*E^((2*Sqrt[-b^2 + 4*a*c]*Sqrt[1 - (4*A
)/(b^2 - 4*a*c)]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[b^2 - 4*a*c])*C[1]
 + 4*a*c*E^((2*Sqrt[-b^2 + 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*ArcTan[(b + 2*c*
x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[b^2 - 4*a*c])*C[1] + b*Sqrt[b^2 - 4*a*c]*Sqrt[1 - (
4*A)/(b^2 - 4*a*c)]*E^((2*Sqrt[-b^2 + 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*ArcTa
n[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[b^2 - 4*a*c])*C[1] + 2*c*Sqrt[b^2 - 4*a*
c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*E^((2*Sqrt[-b^2 + 4*a*c]*Sqrt[1 - (4*A)/(b^2 -
4*a*c)]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[b^2 - 4*a*c])*x*C[1])/(2*(a
 + x*(b + c*x))*(1 + Sqrt[b^2 - 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*E^((2*Sqrt[
-b^2 + 4*a*c]*Sqrt[1 - (4*A)/(b^2 - 4*a*c)]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c
]])/Sqrt[b^2 - 4*a*c])*C[1]))}}

Maple raw input

dsolve((c*x^2+b*x+a)^2*(diff(y(x),x)+y(x)^2)+A = 0, y(x))

Maple raw output

[y(x) = -2*(-I*((I*(4*a*c-b^2)^(1/2)-2*c*x-b)/(2*c*x+b+I*(4*a*c-b^2)^(1/2)))^(-1
/2*c/(-4*a*c+b^2)^(1/2)*(-(4*a*c-b^2+4*A)/c^2)^(1/2))*(-(4*a*c-b^2+4*A)/c^2)^(1/
2)*(4*a*c-b^2)^(1/2)*_C1*c+I*((I*(4*a*c-b^2)^(1/2)-2*c*x-b)/(2*c*x+b+I*(4*a*c-b^
2)^(1/2)))^(1/2*c/(-4*a*c+b^2)^(1/2)*(-(4*a*c-b^2+4*A)/c^2)^(1/2))*(-(4*a*c-b^2+
4*A)/c^2)^(1/2)*(4*a*c-b^2)^(1/2)*c+2*((I*(4*a*c-b^2)^(1/2)-2*c*x-b)/(2*c*x+b+I*
(4*a*c-b^2)^(1/2)))^(-1/2*c/(-4*a*c+b^2)^(1/2)*(-(4*a*c-b^2+4*A)/c^2)^(1/2))*(-4
*a*c+b^2)^(1/2)*_C1*c*x+2*((I*(4*a*c-b^2)^(1/2)-2*c*x-b)/(2*c*x+b+I*(4*a*c-b^2)^
(1/2)))^(1/2*c/(-4*a*c+b^2)^(1/2)*(-(4*a*c-b^2+4*A)/c^2)^(1/2))*(-4*a*c+b^2)^(1/
2)*c*x+((I*(4*a*c-b^2)^(1/2)-2*c*x-b)/(2*c*x+b+I*(4*a*c-b^2)^(1/2)))^(-1/2*c/(-4
*a*c+b^2)^(1/2)*(-(4*a*c-b^2+4*A)/c^2)^(1/2))*(-4*a*c+b^2)^(1/2)*_C1*b+((I*(4*a*
c-b^2)^(1/2)-2*c*x-b)/(2*c*x+b+I*(4*a*c-b^2)^(1/2)))^(1/2*c/(-4*a*c+b^2)^(1/2)*(
-(4*a*c-b^2+4*A)/c^2)^(1/2))*(-4*a*c+b^2)^(1/2)*b)*c/(-4*a*c+b^2)^(1/2)/(2*c*x+b
+I*(4*a*c-b^2)^(1/2))/(I*(4*a*c-b^2)^(1/2)-2*c*x-b)/(_C1*((I*(4*a*c-b^2)^(1/2)-2
*c*x-b)/(2*c*x+b+I*(4*a*c-b^2)^(1/2)))^(-1/2*c/(-4*a*c+b^2)^(1/2)*(-(4*a*c-b^2+4
*A)/c^2)^(1/2))+((I*(4*a*c-b^2)^(1/2)-2*c*x-b)/(2*c*x+b+I*(4*a*c-b^2)^(1/2)))^(1
/2*c/(-4*a*c+b^2)^(1/2)*(-(4*a*c-b^2+4*A)/c^2)^(1/2)))]