##### 4.14.30 $$x^2 (a x+3 y(x))+\left (x^3+y(x)^3\right ) y'(x)=0$$

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

[[_homogeneous, class A], _exact, _rational, _dAlembert]

Book solution method
Exact equation

Mathematica
cpu = 0.515751 (sec), leaf count = 1430

$\left \{\left \{y(x)\to \frac {\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}-\sqrt {-\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}+\frac {\sqrt [3]{3} \left (e^{4 c_1}-a x^4\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}+\sqrt {-\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}+\frac {\sqrt [3]{3} \left (e^{4 c_1}-a x^4\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}+\sqrt {\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}+\frac {\sqrt [3]{3} \left (e^{4 c_1}-a x^4\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}+\frac {\sqrt [3]{3} \left (e^{4 c_1}-a x^4\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}-\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}}\right \}\right \}$

Maple
cpu = 0.232 (sec), leaf count = 29

$\left [y \left (x \right ) = \frac {\RootOf \left (a \,x^{4} \textit {\_C1}^{\frac {4}{3}}+4 x^{3} \textit {\_C1} \textit {\_Z} +\textit {\_Z}^{4}-1\right )}{\textit {\_C1}^{\frac {1}{3}}}\right ]$ Mathematica raw input

DSolve[x^2*(a*x + 3*y[x]) + (x^3 + y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[2
7*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C
[1]) - a*x^4)^3])^(1/3)] - Sqrt[(3^(1/3)*(E^(4*C[1]) - a*x^4))/(9*x^6 + Sqrt[3]*
Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (9*x^6 + Sqrt[3]*Sqrt[27*x^12 +
(E^(4*C[1]) - a*x^4)^3])^(1/3) - (6*Sqrt[2]*x^3)/Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3
^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(
9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)]])/(Sqrt[2]*3^(1/3
))}, {y[x] -> (Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sq
rt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^
(4*C[1]) - a*x^4)^3])^(1/3)] + Sqrt[(3^(1/3)*(E^(4*C[1]) - a*x^4))/(9*x^6 + Sqrt
[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (9*x^6 + Sqrt[3]*Sqrt[27*x^1
2 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (6*Sqrt[2]*x^3)/Sqrt[(-(3^(1/3)*E^(4*C[1]))
 + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3
))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)]])/(Sqrt[2]*3^
(1/3))}, {y[x] -> -((Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt
[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12
 + (E^(4*C[1]) - a*x^4)^3])^(1/3)] + Sqrt[(3^(1/3)*(E^(4*C[1]) - a*x^4))/(9*x^6
+ Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (9*x^6 + Sqrt[3]*Sqrt[
27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) + (6*Sqrt[2]*x^3)/Sqrt[(-(3^(1/3)*E^(4*
C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3]
)^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)]])/(Sqrt
[2]*3^(1/3)))}, {y[x] -> (-Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6
+ Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[2
7*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)] + Sqrt[(3^(1/3)*(E^(4*C[1]) - a*x^4))/(
9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (9*x^6 + Sqrt[3]
*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) + (6*Sqrt[2]*x^3)/Sqrt[(-(3^(1/3)
*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x
^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)]])
/(Sqrt[2]*3^(1/3))}}

Maple raw input

dsolve((x^3+y(x)^3)*diff(y(x),x)+x^2*(a*x+3*y(x)) = 0, y(x))

Maple raw output

[y(x) = RootOf(a*x^4*_C1^(4/3)+4*x^3*_C1*_Z+_Z^4-1)/_C1^(1/3)]