4.41.40 $y^{″}(x){(a+2bx+c{x}^2+y(x{)}^2)}^2+Ay(x)=0$

ODE
$$y^{″}(x){(a+2bx+c{x}^2+y(x{)}^2)}^2+Ay(x)=0$$ ODE Classification

[NONE]

Book solution method
TO DO

Mathematica ✗
cpu = 44.8224 (sec), leaf count = 0 , could not solve

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

Maple ✓
cpu = 1.051 (sec), leaf count = 334

$$\{1(c\arctan \left((cx+b)\frac{1}{\sqrt{ac-b^2}}\right)+{\int }^{y\left(x\right)\frac{1}{\sqrt{c{x}^2+2bx+a}}}\frac{c}{(-ac+b^2){\mathit{\_}\mathit{f}}^4+(\mathit{\_}\mathit{C}\mathit{1}{c}^2-ac+b^2){\mathit{\_}\mathit{f}}^2+\mathit{\_}\mathit{C}\mathit{1}{c}^2+A}\sqrt{({\mathit{\_}\mathit{f}}^2+1)(\mathit{\_}\mathit{C}\mathit{1}({\mathit{\_}\mathit{f}}^2+1)c^2-a{\mathit{\_}\mathit{f}}^2({\mathit{\_}\mathit{f}}^2+1)c+{\mathit{\_}\mathit{f}}^4{b}^2+{\mathit{\_}\mathit{f}}^2{b}^2+A)}d\mathit{\_}\mathit{f}\sqrt{ac-b^2}-\mathit{\_}\mathit{C}\mathit{2}\sqrt{ac-b^2})\frac{1}{\sqrt{ac-b^2}}=0,-1({\int }^{y\left(x\right)\frac{1}{\sqrt{c{x}^2+2bx+a}}}\frac{c}{(-ac+b^2){\mathit{\_}\mathit{f}}^4+(\mathit{\_}\mathit{C}\mathit{1}{c}^2-ac+b^2){\mathit{\_}\mathit{f}}^2+\mathit{\_}\mathit{C}\mathit{1}{c}^2+A}\sqrt{({\mathit{\_}\mathit{f}}^2+1)(\mathit{\_}\mathit{C}\mathit{1}({\mathit{\_}\mathit{f}}^2+1)c^2-a{\mathit{\_}\mathit{f}}^2({\mathit{\_}\mathit{f}}^2+1)c+{\mathit{\_}\mathit{f}}^4{b}^2+{\mathit{\_}\mathit{f}}^2{b}^2+A)}d\mathit{\_}\mathit{f}\sqrt{ac-b^2}+\mathit{\_}\mathit{C}\mathit{2}\sqrt{ac-b^2}-c\arctan \left((cx+b)\frac{1}{\sqrt{ac-b^2}}\right))\frac{1}{\sqrt{ac-b^2}}=0\}$$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

-(Intat(((_f^2+1)*(_C1*(_f^2+1)*c^2-a*_f^2*(_f^2+1)*c+_f^4*b^2+_f^2*b^2+A))^(1/2
)*c/((-a*c+b^2)*_f^4+(_C1*c^2-a*c+b^2)*_f^2+_C1*c^2+A),_f = y(x)/(c*x^2+2*b*x+a)
^(1/2))*(a*c-b^2)^(1/2)+_C2*(a*c-b^2)^(1/2)-c*arctan((c*x+b)/(a*c-b^2)^(1/2)))/(
a*c-b^2)^(1/2) = 0, (c*arctan((c*x+b)/(a*c-b^2)^(1/2))+Intat(((_f^2+1)*(_C1*(_f^
2+1)*c^2-a*_f^2*(_f^2+1)*c+_f^4*b^2+_f^2*b^2+A))^(1/2)*c/((-a*c+b^2)*_f^4+(_C1*c
^2-a*c+b^2)*_f^2+_C1*c^2+A),_f = y(x)/(c*x^2+2*b*x+a)^(1/2))*(a*c-b^2)^(1/2)-_C2
*(a*c-b^2)^(1/2))/(a*c-b^2)^(1/2) = 0