##### 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 = 0.547338 (sec), leaf count = 575

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

Maple
cpu = 0.174 (sec), leaf count = 583

$\left [y \left (x \right ) = \frac {\left (108 \textit {\_C1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 \textit {\_C1}^{4} x^{2}+4 \textit {\_C1} \,x^{4}+18 \textit {\_C1}^{2} x^{2}-x^{2}-4 \textit {\_C1}}\, x \textit {\_C1} +36 x^{2} \textit {\_C1} -8\right )^{\frac {1}{3}}}{6 x \textit {\_C1}}-\frac {2 \left (3 x^{2} \textit {\_C1} -1\right )}{3 \textit {\_C1} x \left (108 \textit {\_C1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 \textit {\_C1}^{4} x^{2}+4 \textit {\_C1} \,x^{4}+18 \textit {\_C1}^{2} x^{2}-x^{2}-4 \textit {\_C1}}\, x \textit {\_C1} +36 x^{2} \textit {\_C1} -8\right )^{\frac {1}{3}}}-\frac {1}{3 x \textit {\_C1}}, y \left (x \right ) = \frac {\left (-12 i x^{2} \textit {\_C1} -i \left (108 \textit {\_C1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 \textit {\_C1}^{4} x^{2}+18 \textit {\_C1}^{2} x^{2}+\left (4 x^{4}-4\right ) \textit {\_C1} -x^{2}}\, x \textit {\_C1} +36 x^{2} \textit {\_C1} -8\right )^{\frac {2}{3}}+4 i\right ) \sqrt {3}+12 x^{2} \textit {\_C1} -\left (\left (108 \textit {\_C1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 \textit {\_C1}^{4} x^{2}+18 \textit {\_C1}^{2} x^{2}+\left (4 x^{4}-4\right ) \textit {\_C1} -x^{2}}\, x \textit {\_C1} +36 x^{2} \textit {\_C1} -8\right )^{\frac {1}{3}}+2\right )^{2}}{12 \left (108 \textit {\_C1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 \textit {\_C1}^{4} x^{2}+18 \textit {\_C1}^{2} x^{2}+\left (4 x^{4}-4\right ) \textit {\_C1} -x^{2}}\, x \textit {\_C1} +36 x^{2} \textit {\_C1} -8\right )^{\frac {1}{3}} x \textit {\_C1}}, y \left (x \right ) = \frac {\left (12 i x^{2} \textit {\_C1} +i \left (108 \textit {\_C1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 \textit {\_C1}^{4} x^{2}+18 \textit {\_C1}^{2} x^{2}+\left (4 x^{4}-4\right ) \textit {\_C1} -x^{2}}\, x \textit {\_C1} +36 x^{2} \textit {\_C1} -8\right )^{\frac {2}{3}}-4 i\right ) \sqrt {3}+12 x^{2} \textit {\_C1} -\left (\left (108 \textit {\_C1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 \textit {\_C1}^{4} x^{2}+18 \textit {\_C1}^{2} x^{2}+\left (4 x^{4}-4\right ) \textit {\_C1} -x^{2}}\, x \textit {\_C1} +36 x^{2} \textit {\_C1} -8\right )^{\frac {1}{3}}+2\right )^{2}}{12 \left (108 \textit {\_C1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 \textit {\_C1}^{4} x^{2}+18 \textit {\_C1}^{2} x^{2}+\left (4 x^{4}-4\right ) \textit {\_C1} -x^{2}}\, x \textit {\_C1} +36 x^{2} \textit {\_C1} -8\right )^{\frac {1}{3}} x \textit {\_C1}}\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*C[1]*(3*x^2 + C[1]))/(C[1]^3 + (9*x^2*(3 + C[1]^2))/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)])/2)^
(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)*(-I + Sqrt[3])*C[1]*(3*x^2 + C[1]))/(C[1]^3 + (9*x^2*(3 + C[1]^2))/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)])/2)^(
1/3) + I*2^(2/3)*(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)}, {y[
x] -> (4*C[1] + ((2*I)*(I + Sqrt[3])*C[1]*(3*x^2 + C[1]))/(C[1]^3 + (9*x^2*(3 +
C[1]^2))/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)])/2)^(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))

Maple raw output

[y(x) = 1/6/x/_C1*(108*_C1^3*x^2+12*3^(1/2)*(27*_C1^4*x^2+4*_C1*x^4+18*_C1^2*x^2
-x^2-4*_C1)^(1/2)*x*_C1+36*x^2*_C1-8)^(1/3)-2/3*(3*_C1*x^2-1)/_C1/x/(108*_C1^3*x
^2+12*3^(1/2)*(27*_C1^4*x^2+4*_C1*x^4+18*_C1^2*x^2-x^2-4*_C1)^(1/2)*x*_C1+36*x^2
*_C1-8)^(1/3)-1/3/x/_C1, y(x) = 1/12*((-12*I*x^2*_C1-I*(108*_C1^3*x^2+12*3^(1/2)
*(27*_C1^4*x^2+18*_C1^2*x^2+(4*x^4-4)*_C1-x^2)^(1/2)*x*_C1+36*x^2*_C1-8)^(2/3)+4
*I)*3^(1/2)+12*x^2*_C1-((108*_C1^3*x^2+12*3^(1/2)*(27*_C1^4*x^2+18*_C1^2*x^2+(4*
x^4-4)*_C1-x^2)^(1/2)*x*_C1+36*x^2*_C1-8)^(1/3)+2)^2)/(108*_C1^3*x^2+12*3^(1/2)*
(27*_C1^4*x^2+18*_C1^2*x^2+(4*x^4-4)*_C1-x^2)^(1/2)*x*_C1+36*x^2*_C1-8)^(1/3)/x/
_C1, y(x) = 1/12*((12*I*x^2*_C1+I*(108*_C1^3*x^2+12*3^(1/2)*(27*_C1^4*x^2+18*_C1
^2*x^2+(4*x^4-4)*_C1-x^2)^(1/2)*x*_C1+36*x^2*_C1-8)^(2/3)-4*I)*3^(1/2)+12*x^2*_C
1-((108*_C1^3*x^2+12*3^(1/2)*(27*_C1^4*x^2+18*_C1^2*x^2+(4*x^4-4)*_C1-x^2)^(1/2)
*x*_C1+36*x^2*_C1-8)^(1/3)+2)^2)/(108*_C1^3*x^2+12*3^(1/2)*(27*_C1^4*x^2+18*_C1^
2*x^2+(4*x^4-4)*_C1-x^2)^(1/2)*x*_C1+36*x^2*_C1-8)^(1/3)/x/_C1]