4.15.7 $y(x)(1-2x^{3}y(x{)}^2)+x(1-2x^{2}y(x{)}^3)y^{\prime }(x)=0$

ODE
$$y(x)(1-2x^{3}y(x{)}^2)+x(1-2x^{2}y(x{)}^3)y^{\prime }(x)=0$$ ODE Classification

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.031638 (sec), leaf count = 672

$$\{\{y(x)\to \frac{c_1{x}^2+\frac{x^4(c_1-2x){}^2}{\sqrt[3]{12c_1{x}^8-6c_1^2{x}^7+c_1^3{x}^6+3\sqrt{3}\sqrt{x^8(-24c_1{x}^4+12c_1^2{x}^3-2c_1^3{x}^2+16x^5+27)}-8x^9-27x^4}}+\sqrt[3]{12c_1{x}^8-6c_1^2{x}^7+c_1^3{x}^6+3\sqrt{3}\sqrt{x^8(-24c_1{x}^4+12c_1^2{x}^3-2c_1^3{x}^2+16x^5+27)}-8x^9-27x^4}-2x^3}{6x^2}\},\{y(x)\to \frac{2x^2(c_1-2x)-\frac{i(\sqrt{3}-i)x^4(c_1-2x){}^2}{\sqrt[3]{12c_1{x}^8-6c_1^2{x}^7+c_1^3{x}^6+3\sqrt{3}\sqrt{x^8(-24c_1{x}^4+12c_1^2{x}^3-2c_1^3{x}^2+16x^5+27)}-8x^9-27x^4}}+i(\sqrt{3}+i)\sqrt[3]{12c_1{x}^8-6c_1^2{x}^7+c_1^3{x}^6+3\sqrt{3}\sqrt{x^8(-24c_1{x}^4+12c_1^2{x}^3-2c_1^3{x}^2+16x^5+27)}-8x^9-27x^4}}{12x^2}\},\{y(x)\to \frac{2x^2(c_1-2x)+\frac{i(\sqrt{3}+i)x^4(c_1-2x){}^2}{\sqrt[3]{12c_1{x}^8-6c_1^2{x}^7+c_1^3{x}^6+3\sqrt{3}\sqrt{x^8(-24c_1{x}^4+12c_1^2{x}^3-2c_1^3{x}^2+16x^5+27)}-8x^9-27x^4}}-(1+i\sqrt{3})\sqrt[3]{12c_1{x}^8-6c_1^2{x}^7+c_1^3{x}^6+3\sqrt{3}\sqrt{x^8(-24c_1{x}^4+12c_1^2{x}^3-2c_1^3{x}^2+16x^5+27)}-8x^9-27x^4}}{12x^2}\}\}$$

Maple ✓
cpu = 0.119 (sec), leaf count = 21

$$\{-2x-\frac{1}{2x^2{\left(y\left(x\right)\right)}^2}-2y\left(x\right)+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[y[x]*(1 - 2*x^3*y[x]^2) + x*(1 - 2*x^2*y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-2*x^3 + x^2*C[1] + (x^4*(-2*x + C[1])^2)/(-27*x^4 - 8*x^9 + 12*x^8*C
[1] - 6*x^7*C[1]^2 + x^6*C[1]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] 
+ 12*x^3*C[1]^2 - 2*x^2*C[1]^3)])^(1/3) + (-27*x^4 - 8*x^9 + 12*x^8*C[1] - 6*x^7
*C[1]^2 + x^6*C[1]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] + 12*x^3*C[
1]^2 - 2*x^2*C[1]^3)])^(1/3))/(6*x^2)}, {y[x] -> (2*x^2*(-2*x + C[1]) - (I*(-I +
 Sqrt[3])*x^4*(-2*x + C[1])^2)/(-27*x^4 - 8*x^9 + 12*x^8*C[1] - 6*x^7*C[1]^2 + x
^6*C[1]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] + 12*x^3*C[1]^2 - 2*x^
2*C[1]^3)])^(1/3) + I*(I + Sqrt[3])*(-27*x^4 - 8*x^9 + 12*x^8*C[1] - 6*x^7*C[1]^
2 + x^6*C[1]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] + 12*x^3*C[1]^2 -
 2*x^2*C[1]^3)])^(1/3))/(12*x^2)}, {y[x] -> (2*x^2*(-2*x + C[1]) + (I*(I + Sqrt[
3])*x^4*(-2*x + C[1])^2)/(-27*x^4 - 8*x^9 + 12*x^8*C[1] - 6*x^7*C[1]^2 + x^6*C[1
]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] + 12*x^3*C[1]^2 - 2*x^2*C[1]
^3)])^(1/3) - (1 + I*Sqrt[3])*(-27*x^4 - 8*x^9 + 12*x^8*C[1] - 6*x^7*C[1]^2 + x^
6*C[1]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] + 12*x^3*C[1]^2 - 2*x^2
*C[1]^3)])^(1/3))/(12*x^2)}}

Maple raw input

dsolve(x*(1-2*x^2*y(x)^3)*diff(y(x),x)+(1-2*x^3*y(x)^2)*y(x) = 0, y(x),'implicit')

Maple raw output

-2*x-1/2/y(x)^2/x^2-2*y(x)+_C1 = 0