##### 4.15.7 $$y(x) \left (1-2 x^3 y(x)^2\right )+x \left (1-2 x^2 y(x)^3\right ) y'(x)=0$$

ODE
$y(x) \left (1-2 x^3 y(x)^2\right )+x \left (1-2 x^2 y(x)^3\right ) y'(x)=0$ ODE Classiﬁcation

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.368218 (sec), leaf count = 672

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

Maple
cpu = 0.142 (sec), leaf count = 770

$\left [y \left (x \right ) = \frac {\left (\left (\textit {\_C1}^{3} x^{2}-6 \textit {\_C1}^{2} x^{3}+12 \textit {\_C1} \,x^{4}-8 x^{5}+3 \sqrt {-6 \textit {\_C1}^{3} x^{2}+36 \textit {\_C1}^{2} x^{3}-72 \textit {\_C1} \,x^{4}+48 x^{5}+81}-27\right ) x \right )^{\frac {1}{3}}}{6 x}+\frac {\left (\textit {\_C1} -2 x \right )^{2} x}{6 \left (\left (\textit {\_C1}^{3} x^{2}-6 \textit {\_C1}^{2} x^{3}+12 \textit {\_C1} \,x^{4}-8 x^{5}+3 \sqrt {-6 \textit {\_C1}^{3} x^{2}+36 \textit {\_C1}^{2} x^{3}-72 \textit {\_C1} \,x^{4}+48 x^{5}+81}-27\right ) x \right )^{\frac {1}{3}}}+\frac {\textit {\_C1}}{6}-\frac {x}{3}, y \left (x \right ) = -\frac {\left (\left (\textit {\_C1}^{3} x^{2}-6 \textit {\_C1}^{2} x^{3}+12 \textit {\_C1} \,x^{4}-8 x^{5}+3 \sqrt {-6 \textit {\_C1}^{3} x^{2}+36 \textit {\_C1}^{2} x^{3}-72 \textit {\_C1} \,x^{4}+48 x^{5}+81}-27\right ) x \right )^{\frac {1}{3}}}{12 x}-\frac {\left (\textit {\_C1} -2 x \right )^{2} x}{12 \left (\left (\textit {\_C1}^{3} x^{2}-6 \textit {\_C1}^{2} x^{3}+12 \textit {\_C1} \,x^{4}-8 x^{5}+3 \sqrt {-6 \textit {\_C1}^{3} x^{2}+36 \textit {\_C1}^{2} x^{3}-72 \textit {\_C1} \,x^{4}+48 x^{5}+81}-27\right ) x \right )^{\frac {1}{3}}}+\frac {\textit {\_C1}}{6}-\frac {x}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (\textit {\_C1}^{3} x^{2}-6 \textit {\_C1}^{2} x^{3}+12 \textit {\_C1} \,x^{4}-8 x^{5}+3 \sqrt {-6 \textit {\_C1}^{3} x^{2}+36 \textit {\_C1}^{2} x^{3}-72 \textit {\_C1} \,x^{4}+48 x^{5}+81}-27\right ) x \right )^{\frac {1}{3}}}{6 x}-\frac {\left (\textit {\_C1} -2 x \right )^{2} x}{6 \left (\left (\textit {\_C1}^{3} x^{2}-6 \textit {\_C1}^{2} x^{3}+12 \textit {\_C1} \,x^{4}-8 x^{5}+3 \sqrt {-6 \textit {\_C1}^{3} x^{2}+36 \textit {\_C1}^{2} x^{3}-72 \textit {\_C1} \,x^{4}+48 x^{5}+81}-27\right ) x \right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (\left (\textit {\_C1}^{3} x^{2}-6 \textit {\_C1}^{2} x^{3}+12 \textit {\_C1} \,x^{4}-8 x^{5}+3 \sqrt {-6 \textit {\_C1}^{3} x^{2}+36 \textit {\_C1}^{2} x^{3}-72 \textit {\_C1} \,x^{4}+48 x^{5}+81}-27\right ) x \right )^{\frac {1}{3}}}{12 x}-\frac {\left (\textit {\_C1} -2 x \right )^{2} x}{12 \left (\left (\textit {\_C1}^{3} x^{2}-6 \textit {\_C1}^{2} x^{3}+12 \textit {\_C1} \,x^{4}-8 x^{5}+3 \sqrt {-6 \textit {\_C1}^{3} x^{2}+36 \textit {\_C1}^{2} x^{3}-72 \textit {\_C1} \,x^{4}+48 x^{5}+81}-27\right ) x \right )^{\frac {1}{3}}}+\frac {\textit {\_C1}}{6}-\frac {x}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (\textit {\_C1}^{3} x^{2}-6 \textit {\_C1}^{2} x^{3}+12 \textit {\_C1} \,x^{4}-8 x^{5}+3 \sqrt {-6 \textit {\_C1}^{3} x^{2}+36 \textit {\_C1}^{2} x^{3}-72 \textit {\_C1} \,x^{4}+48 x^{5}+81}-27\right ) x \right )^{\frac {1}{3}}}{6 x}-\frac {\left (\textit {\_C1} -2 x \right )^{2} x}{6 \left (\left (\textit {\_C1}^{3} x^{2}-6 \textit {\_C1}^{2} x^{3}+12 \textit {\_C1} \,x^{4}-8 x^{5}+3 \sqrt {-6 \textit {\_C1}^{3} x^{2}+36 \textit {\_C1}^{2} x^{3}-72 \textit {\_C1} \,x^{4}+48 x^{5}+81}-27\right ) x \right )^{\frac {1}{3}}}\right )}{2}\right ]$ 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))

Maple raw output

[y(x) = 1/6/x*((_C1^3*x^2-6*_C1^2*x^3+12*_C1*x^4-8*x^5+3*(-6*_C1^3*x^2+36*_C1^2*
x^3-72*_C1*x^4+48*x^5+81)^(1/2)-27)*x)^(1/3)+1/6*(_C1-2*x)^2*x/((_C1^3*x^2-6*_C1
^2*x^3+12*_C1*x^4-8*x^5+3*(-6*_C1^3*x^2+36*_C1^2*x^3-72*_C1*x^4+48*x^5+81)^(1/2)
-27)*x)^(1/3)+1/6*_C1-1/3*x, y(x) = -1/12/x*((_C1^3*x^2-6*_C1^2*x^3+12*_C1*x^4-8
*x^5+3*(-6*_C1^3*x^2+36*_C1^2*x^3-72*_C1*x^4+48*x^5+81)^(1/2)-27)*x)^(1/3)-1/12*
(_C1-2*x)^2*x/((_C1^3*x^2-6*_C1^2*x^3+12*_C1*x^4-8*x^5+3*(-6*_C1^3*x^2+36*_C1^2*
x^3-72*_C1*x^4+48*x^5+81)^(1/2)-27)*x)^(1/3)+1/6*_C1-1/3*x-1/2*I*3^(1/2)*(1/6/x*
((_C1^3*x^2-6*_C1^2*x^3+12*_C1*x^4-8*x^5+3*(-6*_C1^3*x^2+36*_C1^2*x^3-72*_C1*x^4
+48*x^5+81)^(1/2)-27)*x)^(1/3)-1/6*(_C1-2*x)^2*x/((_C1^3*x^2-6*_C1^2*x^3+12*_C1*
x^4-8*x^5+3*(-6*_C1^3*x^2+36*_C1^2*x^3-72*_C1*x^4+48*x^5+81)^(1/2)-27)*x)^(1/3))
, y(x) = -1/12/x*((_C1^3*x^2-6*_C1^2*x^3+12*_C1*x^4-8*x^5+3*(-6*_C1^3*x^2+36*_C1
^2*x^3-72*_C1*x^4+48*x^5+81)^(1/2)-27)*x)^(1/3)-1/12*(_C1-2*x)^2*x/((_C1^3*x^2-6
*_C1^2*x^3+12*_C1*x^4-8*x^5+3*(-6*_C1^3*x^2+36*_C1^2*x^3-72*_C1*x^4+48*x^5+81)^(
1/2)-27)*x)^(1/3)+1/6*_C1-1/3*x+1/2*I*3^(1/2)*(1/6/x*((_C1^3*x^2-6*_C1^2*x^3+12*
_C1*x^4-8*x^5+3*(-6*_C1^3*x^2+36*_C1^2*x^3-72*_C1*x^4+48*x^5+81)^(1/2)-27)*x)^(1
/3)-1/6*(_C1-2*x)^2*x/((_C1^3*x^2-6*_C1^2*x^3+12*_C1*x^4-8*x^5+3*(-6*_C1^3*x^2+3
6*_C1^2*x^3-72*_C1*x^4+48*x^5+81)^(1/2)-27)*x)^(1/3))]