4.14.41 $$4 x^3+9 x^2 y(x)+\left (3 x^3+6 x^2 y(x)-3 x y(x)^2+20 y(x)^3\right ) y'(x)+6 x y(x)^2-y(x)^3=0$$

ODE
$4 x^3+9 x^2 y(x)+\left (3 x^3+6 x^2 y(x)-3 x y(x)^2+20 y(x)^3\right ) y'(x)+6 x y(x)^2-y(x)^3=0$ ODE Classiﬁcation

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

Book solution method
Exact equation

Mathematica
cpu = 0.55555 (sec), leaf count = 2201

$\left \{\left \{y(x)\to \frac {x}{20}+\frac {1}{2} \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}-\frac {1}{2} \sqrt {-\frac {659 x^3}{500 \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}}-\frac {39 x^2}{50}-\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (10 e^{c_1}-13 x^4\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}\right \},\left \{y(x)\to \frac {x}{20}+\frac {1}{2} \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}+\frac {1}{2} \sqrt {-\frac {659 x^3}{500 \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}}-\frac {39 x^2}{50}-\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (10 e^{c_1}-13 x^4\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}\right \},\left \{y(x)\to \frac {x}{20}-\frac {1}{2} \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}-\frac {1}{2} \sqrt {\frac {659 x^3}{500 \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}}-\frac {39 x^2}{50}-\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (10 e^{c_1}-13 x^4\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}\right \},\left \{y(x)\to \frac {x}{20}-\frac {1}{2} \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}+\frac {1}{2} \sqrt {\frac {659 x^3}{500 \sqrt {-\frac {39 x^2}{100}+\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (13 x^4-10 e^{c_1}\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}}-\frac {39 x^2}{50}-\frac {\sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}{5 \sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{\frac {2}{3}} \left (10 e^{c_1}-13 x^4\right )}{5 \sqrt [3]{99 x^6+351 e^{c_1} x^2+\sqrt {3} \sqrt {-67037 x^{12}+185406 e^{c_1} x^8-83733 e^{2 c_1} x^4+32000 e^{3 c_1}}}}}\right \}\right \}$

Maple
cpu = 0.256 (sec), leaf count = 50

$\left [y \left (x \right ) = \frac {\RootOf \left (x^{4} \textit {\_C1}^{4}+3 x^{3} \textit {\_C1}^{3} \textit {\_Z} +3 \textit {\_C1}^{2} \textit {\_Z}^{2} x^{2}-\textit {\_C1} \,\textit {\_Z}^{3} x +5 \textit {\_Z}^{4}-1\right )}{\textit {\_C1}}\right ]$ Mathematica raw input

DSolve[4*x^3 + 9*x^2*y[x] + 6*x*y[x]^2 - y[x]^3 + (3*x^3 + 6*x^2*y[x] - 3*x*y[x]^2 + 20*y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> x/20 + Sqrt[(-39*x^2)/100 + (2*(2/3)^(1/3)*(-10*E^C[1] + 13*x^4))/(5*(
351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 +
 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)) + (351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sq
rt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1
/3)/(5*2^(1/3)*3^(2/3))]/2 - Sqrt[(-39*x^2)/50 + (2*(2/3)^(1/3)*(10*E^C[1] - 13*
x^4))/(5*(351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C
[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)) - (351*E^C[1]*x^2 + 99*x^6 +
Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037
*x^12])^(1/3)/(5*2^(1/3)*3^(2/3)) - (659*x^3)/(500*Sqrt[(-39*x^2)/100 + (2*(2/3)
^(1/3)*(-10*E^C[1] + 13*x^4))/(5*(351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E
^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)) + (35
1*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 1
85406*E^C[1]*x^8 - 67037*x^12])^(1/3)/(5*2^(1/3)*3^(2/3))])]/2}, {y[x] -> x/20 +
 Sqrt[(-39*x^2)/100 + (2*(2/3)^(1/3)*(-10*E^C[1] + 13*x^4))/(5*(351*E^C[1]*x^2 +
 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x
^8 - 67037*x^12])^(1/3)) + (351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[
1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)/(5*2^(1/3)*3
^(2/3))]/2 + Sqrt[(-39*x^2)/50 + (2*(2/3)^(1/3)*(10*E^C[1] - 13*x^4))/(5*(351*E^
C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 18540
6*E^C[1]*x^8 - 67037*x^12])^(1/3)) - (351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[320
00*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)/(5
*2^(1/3)*3^(2/3)) - (659*x^3)/(500*Sqrt[(-39*x^2)/100 + (2*(2/3)^(1/3)*(-10*E^C[
1] + 13*x^4))/(5*(351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 8373
3*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)) + (351*E^C[1]*x^2 + 9
9*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8
 - 67037*x^12])^(1/3)/(5*2^(1/3)*3^(2/3))])]/2}, {y[x] -> x/20 - Sqrt[(-39*x^2)/
100 + (2*(2/3)^(1/3)*(-10*E^C[1] + 13*x^4))/(5*(351*E^C[1]*x^2 + 99*x^6 + Sqrt[3
]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12]
)^(1/3)) + (351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2
*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)/(5*2^(1/3)*3^(2/3))]/2 - Sqr
t[(-39*x^2)/50 + (2*(2/3)^(1/3)*(10*E^C[1] - 13*x^4))/(5*(351*E^C[1]*x^2 + 99*x^
6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 6
7037*x^12])^(1/3)) - (351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) -
83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)/(5*2^(1/3)*3^(2/3)
) + (659*x^3)/(500*Sqrt[(-39*x^2)/100 + (2*(2/3)^(1/3)*(-10*E^C[1] + 13*x^4))/(5
*(351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4
 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)) + (351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*
Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^
(1/3)/(5*2^(1/3)*3^(2/3))])]/2}, {y[x] -> x/20 - Sqrt[(-39*x^2)/100 + (2*(2/3)^(
1/3)*(-10*E^C[1] + 13*x^4))/(5*(351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(
3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)) + (351*
E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185
406*E^C[1]*x^8 - 67037*x^12])^(1/3)/(5*2^(1/3)*3^(2/3))]/2 + Sqrt[(-39*x^2)/50 +
 (2*(2/3)^(1/3)*(10*E^C[1] - 13*x^4))/(5*(351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt
[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3
)) - (351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])
*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)/(5*2^(1/3)*3^(2/3)) + (659*x^3)/(5
00*Sqrt[(-39*x^2)/100 + (2*(2/3)^(1/3)*(-10*E^C[1] + 13*x^4))/(5*(351*E^C[1]*x^2
 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]
*x^8 - 67037*x^12])^(1/3)) + (351*E^C[1]*x^2 + 99*x^6 + Sqrt[3]*Sqrt[32000*E^(3*
C[1]) - 83733*E^(2*C[1])*x^4 + 185406*E^C[1]*x^8 - 67037*x^12])^(1/3)/(5*2^(1/3)
*3^(2/3))])]/2}}

Maple raw input

dsolve((3*x^3+6*x^2*y(x)-3*x*y(x)^2+20*y(x)^3)*diff(y(x),x)+4*x^3+9*x^2*y(x)+6*x*y(x)^2-y(x)^3 = 0, y(x))

Maple raw output

[y(x) = RootOf(_C1^4*x^4+3*_C1^3*_Z*x^3+3*_C1^2*_Z^2*x^2-_C1*_Z^3*x+5*_Z^4-1)/_C
1]