4.15.15 $(2x^{2}y(x{)}^3+xy(x{)}^4+2y(x)+x)y^{\prime }(x)+y(x)(y(x{)}^4+1)=0$

ODE
$$(2x^{2}y(x{)}^3+xy(x{)}^4+2y(x)+x)y^{\prime }(x)+y(x)(y(x{)}^4+1)=0$$ ODE Classification

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.422075 (sec), leaf count = 584

$$\{\{y(x)\to \frac{\frac{2\sqrt[3]{2}{c}_1(c_1+3x^2)}{\sqrt[3]{9(c_1^2+3)x^2+3\sqrt{3}\sqrt{-4c_1^3{x}^6+(-c_1^4+18c_1^2+27)x^4+4c_1^3{x}^2}+2c_1^3}}+2^{2/3}\sqrt[3]{9(c_1^2+3)x^2+3\sqrt{3}\sqrt{-4c_1^3{x}^6+(-c_1^4+18c_1^2+27)x^4+4c_1^3{x}^2}+2c_1^3}+2c_1}{6x}\},\{y(x)\to \frac{-\frac{2i\sqrt[3]{2}(\sqrt{3}-i)c_1(c_1+3x^2)}{\sqrt[3]{9(c_1^2+3)x^2+3\sqrt{3}\sqrt{-4c_1^3{x}^6+(-c_1^4+18c_1^2+27)x^4+4c_1^3{x}^2}+2c_1^3}}+i{2}^{2/3}(\sqrt{3}+i)\sqrt[3]{9(c_1^2+3)x^2+3\sqrt{3}\sqrt{-4c_1^3{x}^6+(-c_1^4+18c_1^2+27)x^4+4c_1^3{x}^2}+2c_1^3}+4c_1}{12x}\},\{y(x)\to \frac{\frac{2i\sqrt[3]{2}(\sqrt{3}+i)c_1(c_1+3x^2)}{\sqrt[3]{9(c_1^2+3)x^2+3\sqrt{3}\sqrt{-4c_1^3{x}^6+(-c_1^4+18c_1^2+27)x^4+4c_1^3{x}^2}+2c_1^3}}-2^{2/3}(1+i\sqrt{3})\sqrt[3]{9(c_1^2+3)x^2+3\sqrt{3}\sqrt{-4c_1^3{x}^6+(-c_1^4+18c_1^2+27)x^4+4c_1^3{x}^2}+2c_1^3}+4c_1}{12x}\}\}$$

Maple ✓
cpu = 0.118 (sec), leaf count = 22

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

DSolve[y[x]*(1 + y[x]^4) + (x + 2*y[x] + 2*x^2*y[x]^3 + x*y[x]^4)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (2*C[1] + (2*2^(1/3)*C[1]*(3*x^2 + C[1]))/(2*C[1]^3 + 9*x^2*(3 + C[1]^
2) + 3*Sqrt[3]*Sqrt[4*x^2*C[1]^3 - 4*x^6*C[1]^3 + x^4*(27 + 18*C[1]^2 - C[1]^4)]
)^(1/3) + 2^(2/3)*(2*C[1]^3 + 9*x^2*(3 + C[1]^2) + 3*Sqrt[3]*Sqrt[4*x^2*C[1]^3 -
 4*x^6*C[1]^3 + x^4*(27 + 18*C[1]^2 - C[1]^4)])^(1/3))/(6*x)}, {y[x] -> (4*C[1] 
- ((2*I)*2^(1/3)*(-I + Sqrt[3])*C[1]*(3*x^2 + C[1]))/(2*C[1]^3 + 9*x^2*(3 + C[1]
^2) + 3*Sqrt[3]*Sqrt[4*x^2*C[1]^3 - 4*x^6*C[1]^3 + x^4*(27 + 18*C[1]^2 - C[1]^4)
])^(1/3) + I*2^(2/3)*(I + Sqrt[3])*(2*C[1]^3 + 9*x^2*(3 + C[1]^2) + 3*Sqrt[3]*Sq
rt[4*x^2*C[1]^3 - 4*x^6*C[1]^3 + x^4*(27 + 18*C[1]^2 - C[1]^4)])^(1/3))/(12*x)},
 {y[x] -> (4*C[1] + ((2*I)*2^(1/3)*(I + Sqrt[3])*C[1]*(3*x^2 + C[1]))/(2*C[1]^3 
+ 9*x^2*(3 + C[1]^2) + 3*Sqrt[3]*Sqrt[4*x^2*C[1]^3 - 4*x^6*C[1]^3 + x^4*(27 + 18
*C[1]^2 - C[1]^4)])^(1/3) - 2^(2/3)*(1 + I*Sqrt[3])*(2*C[1]^3 + 9*x^2*(3 + C[1]^
2) + 3*Sqrt[3]*Sqrt[4*x^2*C[1]^3 - 4*x^6*C[1]^3 + x^4*(27 + 18*C[1]^2 - C[1]^4)]
)^(1/3))/(12*x)}}

Maple raw input

dsolve((x+2*y(x)+2*x^2*y(x)^3+x*y(x)^4)*diff(y(x),x)+(1+y(x)^4)*y(x) = 0, y(x),'implicit')

Maple raw output

_C1+(x+y(x))*y(x)/(x*y(x)^3-1) = 0