4.13.28 $$3 y(x)^2 y'(x)=a y(x)^3+x+1$$

ODE
$3 y(x)^2 y'(x)=a y(x)^3+x+1$ ODE Classiﬁcation

[_rational, _Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.319325 (sec), leaf count = 111

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

Maple
cpu = 0.049 (sec), leaf count = 154

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

DSolve[3*y[x]^2*y'[x] == 1 + x + a*y[x]^3,y[x],x]

Mathematica raw output

{{y[x] -> (-1 - a*(1 + x) + a^2*E^(a*x)*C[1])^(1/3)/a^(2/3)}, {y[x] -> -(((-1)^(
1/3)*(-1 - a*(1 + x) + a^2*E^(a*x)*C[1])^(1/3))/a^(2/3))}, {y[x] -> ((-1)^(2/3)*
(-1 - a*(1 + x) + a^2*E^(a*x)*C[1])^(1/3))/a^(2/3)}}

Maple raw input

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

Maple raw output

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