##### 4.12.36 $$\left (1-x^3 y(x)\right ) y'(x)=x^2 y(x)^2$$

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

[[_homogeneous, class G], _rational, [_Abel, 2nd type, class B]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 11.0687 (sec), leaf count = 326

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

Maple
cpu = 0.586 (sec), leaf count = 789

$\left [y \left (x \right ) = \frac {\left (\frac {\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}-\frac {\textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )^{2}+3}{2 x^{3}}, y \left (x \right ) = \frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} \left (\frac {\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}-\frac {\textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )^{2}+3}{2 x^{3}}, y \left (x \right ) = \frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} \left (\frac {\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}-\frac {\textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )^{2}+3}{2 x^{3}}, y \left (x \right ) = \frac {\frac {\left (-\frac {4 \left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {4 \textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {\textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64}+3}{2 x^{3}}, y \left (x \right ) = \frac {\frac {\left (-\frac {4 \left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {4 \textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {\textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64}+3}{2 x^{3}}, y \left (x \right ) = \frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} \left (-\frac {4 \left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {4 \textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {\textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64}+3}{2 x^{3}}, y \left (x \right ) = \frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} \left (-\frac {4 \left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {4 \textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {\textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64}+3}{2 x^{3}}, y \left (x \right ) = \frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} \left (-\frac {4 \left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {4 \textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {\textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64}+3}{2 x^{3}}, y \left (x \right ) = \frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} \left (-\frac {4 \left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {4 \textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{\textit {\_C1}}+\frac {\textit {\_C1}}{\left (x^{3}+\sqrt {\textit {\_C1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64}+3}{2 x^{3}}\right ]$ Mathematica raw input

DSolve[(1 - x^3*y[x])*y'[x] == x^2*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> (1 + (1 + 12*x^6*C[1] + 2*Sqrt[6]*Sqrt[x^6*C[1]*(1 + 6*x^6*C[1])])^(-1
/3) + (1 + 12*x^6*C[1] + 2*Sqrt[6]*Sqrt[x^6*C[1]*(1 + 6*x^6*C[1])])^(1/3))/(2*x^
3)}, {y[x] -> (4 - (2*(1 + I*Sqrt[3]))/(1 + 12*x^6*C[1] + 2*Sqrt[6]*Sqrt[x^6*C[1
]*(1 + 6*x^6*C[1])])^(1/3) + (2*I)*(I + Sqrt[3])*(1 + 12*x^6*C[1] + 2*Sqrt[6]*Sq
rt[x^6*C[1]*(1 + 6*x^6*C[1])])^(1/3))/(8*x^3)}, {y[x] -> (4 + ((2*I)*(I + Sqrt[3
]))/(1 + 12*x^6*C[1] + 2*Sqrt[6]*Sqrt[x^6*C[1]*(1 + 6*x^6*C[1])])^(1/3) - 2*(1 +
 I*Sqrt[3])*(1 + 12*x^6*C[1] + 2*Sqrt[6]*Sqrt[x^6*C[1]*(1 + 6*x^6*C[1])])^(1/3))
/(8*x^3)}}

Maple raw input

dsolve((1-x^3*y(x))*diff(y(x),x) = x^2*y(x)^2, y(x))

Maple raw output

[y(x) = 1/2*((1/_C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3)-_C1/(x^3+(_C1^6+x^6)^(1/2))^(1
/3))^2+3)/x^3, y(x) = 1/2*((-1/2-1/2*I*3^(1/2))^6*(1/_C1*(x^3+(_C1^6+x^6)^(1/2))
^(1/3)-_C1/(x^3+(_C1^6+x^6)^(1/2))^(1/3))^2+3)/x^3, y(x) = 1/2*((-1/2+1/2*I*3^(1
/2))^6*(1/_C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3)-_C1/(x^3+(_C1^6+x^6)^(1/2))^(1/3))^2
+3)/x^3, y(x) = 1/2*(1/64*(-4/_C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3)+4*_C1/(x^3+(_C1^
6+x^6)^(1/2))^(1/3)-4*I*3^(1/2)*(1/_C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3)+_C1/(x^3+(_
C1^6+x^6)^(1/2))^(1/3)))^2+3)/x^3, y(x) = 1/2*(1/64*(-4/_C1*(x^3+(_C1^6+x^6)^(1/
2))^(1/3)+4*_C1/(x^3+(_C1^6+x^6)^(1/2))^(1/3)+4*I*3^(1/2)*(1/_C1*(x^3+(_C1^6+x^6
)^(1/2))^(1/3)+_C1/(x^3+(_C1^6+x^6)^(1/2))^(1/3)))^2+3)/x^3, y(x) = 1/2*(1/64*(-
1/2-1/2*I*3^(1/2))^6*(-4/_C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3)+4*_C1/(x^3+(_C1^6+x^6
)^(1/2))^(1/3)-4*I*3^(1/2)*(1/_C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3)+_C1/(x^3+(_C1^6+
x^6)^(1/2))^(1/3)))^2+3)/x^3, y(x) = 1/2*(1/64*(-1/2-1/2*I*3^(1/2))^6*(-4/_C1*(x
^3+(_C1^6+x^6)^(1/2))^(1/3)+4*_C1/(x^3+(_C1^6+x^6)^(1/2))^(1/3)+4*I*3^(1/2)*(1/_
C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3)+_C1/(x^3+(_C1^6+x^6)^(1/2))^(1/3)))^2+3)/x^3, y
(x) = 1/2*(1/64*(-1/2+1/2*I*3^(1/2))^6*(-4/_C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3)+4*_
C1/(x^3+(_C1^6+x^6)^(1/2))^(1/3)-4*I*3^(1/2)*(1/_C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3
)+_C1/(x^3+(_C1^6+x^6)^(1/2))^(1/3)))^2+3)/x^3, y(x) = 1/2*(1/64*(-1/2+1/2*I*3^(
1/2))^6*(-4/_C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3)+4*_C1/(x^3+(_C1^6+x^6)^(1/2))^(1/3
)+4*I*3^(1/2)*(1/_C1*(x^3+(_C1^6+x^6)^(1/2))^(1/3)+_C1/(x^3+(_C1^6+x^6)^(1/2))^(
1/3)))^2+3)/x^3]