4.14.40 $2x^3+3x^{2}y(x)+(-x^3+x^2+3xy(x{)}^2+2y(x{)}^3)y^{\prime }(x)-y(x{)}^3+y(x{)}^2=0$

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

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.0442185 (sec), leaf count = 364

$$\{\{y(x)\to \frac{6\sqrt[3]{2}(c_1+x)-2^{2/3}(\sqrt{729(c_{1}x+x^3){}^2+108(c_1+x){}^3}+27c_{1}x+27x^3){}^{2/3}}{6\sqrt[3]{\sqrt{729(c_{1}x+x^3){}^2+108(c_1+x){}^3}+27c_{1}x+27x^3}}\},\{y(x)\to \frac{2^{2/3}(1-i\sqrt{3})(\sqrt{729(c_{1}x+x^3){}^2+108(c_1+x){}^3}+27c_{1}x+27x^3){}^{2/3}-6i\sqrt[3]{2}(\sqrt{3}-i)(c_1+x)}{12\sqrt[3]{\sqrt{729(c_{1}x+x^3){}^2+108(c_1+x){}^3}+27c_{1}x+27x^3}}\},\{y(x)\to \frac{2^{2/3}(1+i\sqrt{3})(\sqrt{729(c_{1}x+x^3){}^2+108(c_1+x){}^3}+27c_{1}x+27x^3){}^{2/3}+6i\sqrt[3]{2}(\sqrt{3}+i)(c_1+x)}{12\sqrt[3]{\sqrt{729(c_{1}x+x^3){}^2+108(c_1+x){}^3}+27c_{1}x+27x^3}}\}\}$$

Maple ✓
cpu = 0.162 (sec), leaf count = 28

$$\{\frac{{\left(y\left(x\right)\right)}^3+(x+\mathit{\_}\mathit{C}\mathit{1})y\left(x\right)+x(x^2+\mathit{\_}\mathit{C}\mathit{1})}{x+y\left(x\right)}=0\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (6*2^(1/3)*(x + C[1]) - 2^(2/3)*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[
1])^3 + 729*(x^3 + x*C[1])^2])^(2/3))/(6*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1
])^3 + 729*(x^3 + x*C[1])^2])^(1/3))}, {y[x] -> ((-6*I)*2^(1/3)*(-I + Sqrt[3])*(
x + C[1]) + 2^(2/3)*(1 - I*Sqrt[3])*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 
+ 729*(x^3 + x*C[1])^2])^(2/3))/(12*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 
+ 729*(x^3 + x*C[1])^2])^(1/3))}, {y[x] -> ((6*I)*2^(1/3)*(I + Sqrt[3])*(x + C[1
]) + 2^(2/3)*(1 + I*Sqrt[3])*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 + 729*(
x^3 + x*C[1])^2])^(2/3))/(12*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 + 729*(
x^3 + x*C[1])^2])^(1/3))}}

Maple raw input

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

Maple raw output

(y(x)^3+(x+_C1)*y(x)+x*(x^2+_C1))/(x+y(x)) = 0