4.8.30 $(y^{\prime }(x)+y(x{)}^2){(a+bx+c{x}^2)}^2+A=0$

ODE
$$(y^{\prime }(x)+y(x{)}^2){(a+bx+c{x}^2)}^2+A=0$$ ODE Classification

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica ✓
cpu = 1.46923 (sec), leaf count = 612

$$\left\{\{y(x)\to \frac{b^2{c}_1(-\exp \left(\frac{2\sqrt{4ac-b^2}\sqrt{1-\frac{4A}{b^2-4ac}}{\tan }^{-1}\left(\frac{b+2cx}{\sqrt{4ac-b^2}}\right)}{\sqrt{b^2-4ac}}\right))+b{c}_1\sqrt{b^2-4ac}\sqrt{1-\frac{4A}{b^2-4ac}}\exp \left(\frac{2\sqrt{4ac-b^2}\sqrt{1-\frac{4A}{b^2-4ac}}{\tan }^{-1}\left(\frac{b+2cx}{\sqrt{4ac-b^2}}\right)}{\sqrt{b^2-4ac}}\right)+4A{c}_1\exp \left(\frac{2\sqrt{4ac-b^2}\sqrt{1-\frac{4A}{b^2-4ac}}{\tan }^{-1}\left(\frac{b+2cx}{\sqrt{4ac-b^2}}\right)}{\sqrt{b^2-4ac}}\right)+4ac{c}_1\exp \left(\frac{2\sqrt{4ac-b^2}\sqrt{1-\frac{4A}{b^2-4ac}}{\tan }^{-1}\left(\frac{b+2cx}{\sqrt{4ac-b^2}}\right)}{\sqrt{b^2-4ac}}\right)+2c{c}_{1}x\sqrt{b^2-4ac}\sqrt{1-\frac{4A}{b^2-4ac}}\exp \left(\frac{2\sqrt{4ac-b^2}\sqrt{1-\frac{4A}{b^2-4ac}}{\tan }^{-1}\left(\frac{b+2cx}{\sqrt{4ac-b^2}}\right)}{\sqrt{b^2-4ac}}\right)+\sqrt{b^2-4ac}\sqrt{1-\frac{4A}{b^2-4ac}}+b+2cx}{2(a+x(b+cx))(c_1\sqrt{b^2-4ac}\sqrt{1-\frac{4A}{b^2-4ac}}\exp \left(\frac{2\sqrt{4ac-b^2}\sqrt{1-\frac{4A}{b^2-4ac}}{\tan }^{-1}\left(\frac{b+2cx}{\sqrt{4ac-b^2}}\right)}{\sqrt{b^2-4ac}}\right)+1)}\}\right\}$$

Maple ✓
cpu = 0.383 (sec), leaf count = 490

$$\{y\left(x\right)=2\frac{c}{\sqrt{-4ca+b^2}(2cx+b+i\sqrt{4ca-b^2})(i\sqrt{4ca-b^2}-2cx-b)}((i\sqrt{\frac{-4ca+b^2-4A}{c^2}}c\sqrt{4ca-b^2}-\sqrt{-4ca+b^2}(2cx+b))\mathit{\_}\mathit{C}\mathit{1}{\left(\frac{i\sqrt{4ca-b^2}-2cx-b}{2cx+b+i\sqrt{4ca-b^2}}\right)}^{-1/2\frac{c}{\sqrt{-4ca+b^2}}\sqrt{\frac{-4ca+b^2-4A}{c^2}}}-{\left(\frac{i\sqrt{4ca-b^2}-2cx-b}{2cx+b+i\sqrt{4ca-b^2}}\right)}^{1/2\frac{c}{\sqrt{-4ca+b^2}}\sqrt{\frac{-4ca+b^2-4A}{c^2}}}(i\sqrt{\frac{-4ca+b^2-4A}{c^2}}c\sqrt{4ca-b^2}+\sqrt{-4ca+b^2}(2cx+b))){(\mathit{\_}\mathit{C}\mathit{1}{\left(\frac{i\sqrt{4ca-b^2}-2cx-b}{2cx+b+i\sqrt{4ca-b^2}}\right)}^{-1/2\frac{c}{\sqrt{-4ca+b^2}}\sqrt{\frac{-4ca+b^2-4A}{c^2}}}+{\left(\frac{i\sqrt{4ca-b^2}-2cx-b}{2cx+b+i\sqrt{4ca-b^2}}\right)}^{1/2\frac{c}{\sqrt{-4ca+b^2}}\sqrt{\frac{-4ca+b^2-4A}{c^2}}})}^{-1}\}$$ 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),'implicit')

Maple raw output

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