##### 4.12.7 $$x (3 y(x)+2 x) y'(x)=y(x)^2$$

ODE
$x (3 y(x)+2 x) y'(x)=y(x)^2$ ODE Classiﬁcation

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.387938 (sec), leaf count = 413

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

Maple
cpu = 0.202 (sec), leaf count = 461

$\left [y \left (x \right ) = \frac {\frac {\left (108 x \textit {\_C1} -8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{4} \textit {\_C1}^{4}+81 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 \textit {\_C1}^{2} x^{2}}{3 \left (108 x \textit {\_C1} -8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{4} \textit {\_C1}^{4}+81 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}-\frac {x \textit {\_C1}}{3}}{\textit {\_C1}}, y \left (x \right ) = \frac {-\frac {\left (108 x \textit {\_C1} -8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{4} \textit {\_C1}^{4}+81 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}{12}-\frac {\textit {\_C1}^{2} x^{2}}{3 \left (108 x \textit {\_C1} -8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{4} \textit {\_C1}^{4}+81 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}-\frac {x \textit {\_C1}}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (108 x \textit {\_C1} -8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{4} \textit {\_C1}^{4}+81 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 \textit {\_C1}^{2} x^{2}}{3 \left (108 x \textit {\_C1} -8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{4} \textit {\_C1}^{4}+81 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}\right )}{2}}{\textit {\_C1}}, y \left (x \right ) = \frac {-\frac {\left (108 x \textit {\_C1} -8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{4} \textit {\_C1}^{4}+81 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}{12}-\frac {\textit {\_C1}^{2} x^{2}}{3 \left (108 x \textit {\_C1} -8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{4} \textit {\_C1}^{4}+81 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}-\frac {x \textit {\_C1}}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (108 x \textit {\_C1} -8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{4} \textit {\_C1}^{4}+81 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 \textit {\_C1}^{2} x^{2}}{3 \left (108 x \textit {\_C1} -8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{4} \textit {\_C1}^{4}+81 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}\right )}{2}}{\textit {\_C1}}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = 1/_C1*(1/6*(108*x*_C1-8*x^3*_C1^3+12*(-12*_C1^4*x^4+81*_C1^2*x^2)^(1/2))
^(1/3)+2/3*_C1^2*x^2/(108*x*_C1-8*x^3*_C1^3+12*(-12*_C1^4*x^4+81*_C1^2*x^2)^(1/2
))^(1/3)-1/3*x*_C1), y(x) = 1/_C1*(-1/12*(108*x*_C1-8*x^3*_C1^3+12*(-12*_C1^4*x^
4+81*_C1^2*x^2)^(1/2))^(1/3)-1/3*_C1^2*x^2/(108*x*_C1-8*x^3*_C1^3+12*(-12*_C1^4*
x^4+81*_C1^2*x^2)^(1/2))^(1/3)-1/3*x*_C1-1/2*I*3^(1/2)*(1/6*(108*x*_C1-8*x^3*_C1
^3+12*(-12*_C1^4*x^4+81*_C1^2*x^2)^(1/2))^(1/3)-2/3*_C1^2*x^2/(108*x*_C1-8*x^3*_
C1^3+12*(-12*_C1^4*x^4+81*_C1^2*x^2)^(1/2))^(1/3))), y(x) = 1/_C1*(-1/12*(108*x*
_C1-8*x^3*_C1^3+12*(-12*_C1^4*x^4+81*_C1^2*x^2)^(1/2))^(1/3)-1/3*_C1^2*x^2/(108*
x*_C1-8*x^3*_C1^3+12*(-12*_C1^4*x^4+81*_C1^2*x^2)^(1/2))^(1/3)-1/3*x*_C1+1/2*I*3
^(1/2)*(1/6*(108*x*_C1-8*x^3*_C1^3+12*(-12*_C1^4*x^4+81*_C1^2*x^2)^(1/2))^(1/3)-
2/3*_C1^2*x^2/(108*x*_C1-8*x^3*_C1^3+12*(-12*_C1^4*x^4+81*_C1^2*x^2)^(1/2))^(1/3
)))]