##### 4.13.31 $$3 \left (x^2-y(x)^2\right ) y'(x)-2 y(x)^3+6 x (x+1) y(x)+3 e^x=0$$

ODE
$3 \left (x^2-y(x)^2\right ) y'(x)-2 y(x)^3+6 x (x+1) y(x)+3 e^x=0$ ODE Classiﬁcation

[y=_G(x,y')]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.645496 (sec), leaf count = 497

$\left \{\left \{y(x)\to -\frac {e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} e^{2 x} x^2}{\sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}{2 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) e^{2 x} x^2}{2^{2/3} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}{2 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) e^{2 x} x^2}{2^{2/3} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}\right \}\right \}$

Maple
cpu = 0.082 (sec), leaf count = 622

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

DSolve[3*E^x + 6*x*(1 + x)*y[x] - 2*y[x]^3 + 3*(x^2 - y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((2^(1/3)*E^(2*x)*x^2)/(-E^(7*x) + E^(4*x)*C[1] + Sqrt[E^(8*x)*(E^(6*
x) - 4*E^(4*x)*x^6 - 2*E^(3*x)*C[1] + C[1]^2)])^(1/3)) - (-E^(7*x) + E^(4*x)*C[1
] + Sqrt[E^(8*x)*(E^(6*x) - 4*E^(4*x)*x^6 - 2*E^(3*x)*C[1] + C[1]^2)])^(1/3)/(2^
(1/3)*E^(2*x))}, {y[x] -> ((1 + I*Sqrt[3])*E^(2*x)*x^2)/(2^(2/3)*(-E^(7*x) + E^(
4*x)*C[1] + Sqrt[E^(8*x)*(E^(6*x) - 4*E^(4*x)*x^6 - 2*E^(3*x)*C[1] + C[1]^2)])^(
1/3)) + ((1 - I*Sqrt[3])*(-E^(7*x) + E^(4*x)*C[1] + Sqrt[E^(8*x)*(E^(6*x) - 4*E^
(4*x)*x^6 - 2*E^(3*x)*C[1] + C[1]^2)])^(1/3))/(2*2^(1/3)*E^(2*x))}, {y[x] -> ((1
 - I*Sqrt[3])*E^(2*x)*x^2)/(2^(2/3)*(-E^(7*x) + E^(4*x)*C[1] + Sqrt[E^(8*x)*(E^(
6*x) - 4*E^(4*x)*x^6 - 2*E^(3*x)*C[1] + C[1]^2)])^(1/3)) + ((1 + I*Sqrt[3])*(-E^
(7*x) + E^(4*x)*C[1] + Sqrt[E^(8*x)*(E^(6*x) - 4*E^(4*x)*x^6 - 2*E^(3*x)*C[1] +
C[1]^2)])^(1/3))/(2*2^(1/3)*E^(2*x))}}

Maple raw input

dsolve(3*(x^2-y(x)^2)*diff(y(x),x)+3*exp(x)+6*x*y(x)*(x+1)-2*y(x)^3 = 0, y(x))

Maple raw output

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