4.15.15 \(\left (2 x^2 y(x)^3+x y(x)^4+2 y(x)+x\right ) y'(x)+y(x) \left (y(x)^4+1\right )=0\)

ODE
\[ \left (2 x^2 y(x)^3+x y(x)^4+2 y(x)+x\right ) y'(x)+y(x) \left (y(x)^4+1\right )=0 \] ODE Classification

[_rational]

Book solution method
Exact equation, integrating factor

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

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

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

\[ \left \{ {\it \_C1}+{\frac { \left (x+y \relax (x ) \right ) y \relax (x ) }{x \left (y \relax (x ) \right ) ^{3}-1}}=0 \right \} \] 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