##### 4.13.34 $$\left (-3 x^2 y(x)+6 y(x)^2+1\right ) y'(x)+x^2-3 x y(x)^2=0$$

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

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.481151 (sec), leaf count = 570

$\left \{\left \{y(x)\to \frac {x^2}{4}-\frac {\sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}{6 \sqrt [3]{2}}+\frac {6-\frac {9 x^4}{4}}{3\ 2^{2/3} \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}\right \},\left \{y(x)\to \frac {x^2}{4}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}-\frac {\left (1+i \sqrt {3}\right ) \left (6-\frac {9 x^4}{4}\right )}{6\ 2^{2/3} \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}\right \},\left \{y(x)\to \frac {x^2}{4}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}-\frac {\left (1-i \sqrt {3}\right ) \left (6-\frac {9 x^4}{4}\right )}{6\ 2^{2/3} \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}\right \}\right \}$

Maple
cpu = 0.043 (sec), leaf count = 797

$\left [y \left (x \right ) = \frac {\left (-108 x^{2}-144 x^{3}-432 \textit {\_C1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 \textit {\_C1} \,x^{6}+144 x^{6}+216 x^{5}+864 \textit {\_C1} \,x^{3}-27 x^{4}+648 x^{2} \textit {\_C1} +1296 \textit {\_C1}^{2}+96}\right )^{\frac {1}{3}}}{12}-\frac {12 \left (\frac {1}{6}-\frac {x^{4}}{16}\right )}{\left (-108 x^{2}-144 x^{3}-432 \textit {\_C1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 \textit {\_C1} \,x^{6}+144 x^{6}+216 x^{5}+864 \textit {\_C1} \,x^{3}-27 x^{4}+648 x^{2} \textit {\_C1} +1296 \textit {\_C1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4}, y \left (x \right ) = -\frac {\left (-108 x^{2}-144 x^{3}-432 \textit {\_C1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 \textit {\_C1} \,x^{6}+144 x^{6}+216 x^{5}+864 \textit {\_C1} \,x^{3}-27 x^{4}+648 x^{2} \textit {\_C1} +1296 \textit {\_C1}^{2}+96}\right )^{\frac {1}{3}}}{24}+\frac {1-\frac {3 x^{4}}{8}}{\left (-108 x^{2}-144 x^{3}-432 \textit {\_C1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 \textit {\_C1} \,x^{6}+144 x^{6}+216 x^{5}+864 \textit {\_C1} \,x^{3}-27 x^{4}+648 x^{2} \textit {\_C1} +1296 \textit {\_C1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{2}-144 x^{3}-432 \textit {\_C1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 \textit {\_C1} \,x^{6}+144 x^{6}+216 x^{5}+864 \textit {\_C1} \,x^{3}-27 x^{4}+648 x^{2} \textit {\_C1} +1296 \textit {\_C1}^{2}+96}\right )^{\frac {1}{3}}}{12}+\frac {2-\frac {3 x^{4}}{4}}{\left (-108 x^{2}-144 x^{3}-432 \textit {\_C1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 \textit {\_C1} \,x^{6}+144 x^{6}+216 x^{5}+864 \textit {\_C1} \,x^{3}-27 x^{4}+648 x^{2} \textit {\_C1} +1296 \textit {\_C1}^{2}+96}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (-108 x^{2}-144 x^{3}-432 \textit {\_C1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 \textit {\_C1} \,x^{6}+144 x^{6}+216 x^{5}+864 \textit {\_C1} \,x^{3}-27 x^{4}+648 x^{2} \textit {\_C1} +1296 \textit {\_C1}^{2}+96}\right )^{\frac {1}{3}}}{24}+\frac {1-\frac {3 x^{4}}{8}}{\left (-108 x^{2}-144 x^{3}-432 \textit {\_C1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 \textit {\_C1} \,x^{6}+144 x^{6}+216 x^{5}+864 \textit {\_C1} \,x^{3}-27 x^{4}+648 x^{2} \textit {\_C1} +1296 \textit {\_C1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{2}-144 x^{3}-432 \textit {\_C1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 \textit {\_C1} \,x^{6}+144 x^{6}+216 x^{5}+864 \textit {\_C1} \,x^{3}-27 x^{4}+648 x^{2} \textit {\_C1} +1296 \textit {\_C1}^{2}+96}\right )^{\frac {1}{3}}}{12}+\frac {2-\frac {3 x^{4}}{4}}{\left (-108 x^{2}-144 x^{3}-432 \textit {\_C1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 \textit {\_C1} \,x^{6}+144 x^{6}+216 x^{5}+864 \textit {\_C1} \,x^{3}-27 x^{4}+648 x^{2} \textit {\_C1} +1296 \textit {\_C1}^{2}+96}\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> x^2/4 + (6 - (9*x^4)/4)/(3*2^(2/3)*(27*x^2 + 36*x^3 - (27*x^6)/4 + 108
*C[1] + Sqrt[4*(6 - (9*x^4)/4)^3 + (27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1])^2])
^(1/3)) - (27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1] + Sqrt[4*(6 - (9*x^4)/4)^3 +
(27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1])^2])^(1/3)/(6*2^(1/3))}, {y[x] -> x^2/4
 - ((1 + I*Sqrt[3])*(6 - (9*x^4)/4))/(6*2^(2/3)*(27*x^2 + 36*x^3 - (27*x^6)/4 +
108*C[1] + Sqrt[4*(6 - (9*x^4)/4)^3 + (27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1])^
2])^(1/3)) + ((1 - I*Sqrt[3])*(27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1] + Sqrt[4*
(6 - (9*x^4)/4)^3 + (27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1])^2])^(1/3))/(12*2^(
1/3))}, {y[x] -> x^2/4 - ((1 - I*Sqrt[3])*(6 - (9*x^4)/4))/(6*2^(2/3)*(27*x^2 +
36*x^3 - (27*x^6)/4 + 108*C[1] + Sqrt[4*(6 - (9*x^4)/4)^3 + (27*x^2 + 36*x^3 - (
27*x^6)/4 + 108*C[1])^2])^(1/3)) + ((1 + I*Sqrt[3])*(27*x^2 + 36*x^3 - (27*x^6)/
4 + 108*C[1] + Sqrt[4*(6 - (9*x^4)/4)^3 + (27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[
1])^2])^(1/3))/(12*2^(1/3))}}

Maple raw input

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

Maple raw output

[y(x) = 1/12*(-108*x^2-144*x^3-432*_C1+27*x^6+12*(-54*x^9-162*_C1*x^6+144*x^6+21
6*x^5+864*_C1*x^3-27*x^4+648*_C1*x^2+1296*_C1^2+96)^(1/2))^(1/3)-12*(1/6-1/16*x^
4)/(-108*x^2-144*x^3-432*_C1+27*x^6+12*(-54*x^9-162*_C1*x^6+144*x^6+216*x^5+864*
_C1*x^3-27*x^4+648*_C1*x^2+1296*_C1^2+96)^(1/2))^(1/3)+1/4*x^2, y(x) = -1/24*(-1
08*x^2-144*x^3-432*_C1+27*x^6+12*(-54*x^9-162*_C1*x^6+144*x^6+216*x^5+864*_C1*x^
3-27*x^4+648*_C1*x^2+1296*_C1^2+96)^(1/2))^(1/3)+6*(1/6-1/16*x^4)/(-108*x^2-144*
x^3-432*_C1+27*x^6+12*(-54*x^9-162*_C1*x^6+144*x^6+216*x^5+864*_C1*x^3-27*x^4+64
8*_C1*x^2+1296*_C1^2+96)^(1/2))^(1/3)+1/4*x^2-1/2*I*3^(1/2)*(1/12*(-108*x^2-144*
x^3-432*_C1+27*x^6+12*(-54*x^9-162*_C1*x^6+144*x^6+216*x^5+864*_C1*x^3-27*x^4+64
8*_C1*x^2+1296*_C1^2+96)^(1/2))^(1/3)+12*(1/6-1/16*x^4)/(-108*x^2-144*x^3-432*_C
1+27*x^6+12*(-54*x^9-162*_C1*x^6+144*x^6+216*x^5+864*_C1*x^3-27*x^4+648*_C1*x^2+
1296*_C1^2+96)^(1/2))^(1/3)), y(x) = -1/24*(-108*x^2-144*x^3-432*_C1+27*x^6+12*(
-54*x^9-162*_C1*x^6+144*x^6+216*x^5+864*_C1*x^3-27*x^4+648*_C1*x^2+1296*_C1^2+96
)^(1/2))^(1/3)+6*(1/6-1/16*x^4)/(-108*x^2-144*x^3-432*_C1+27*x^6+12*(-54*x^9-162
*_C1*x^6+144*x^6+216*x^5+864*_C1*x^3-27*x^4+648*_C1*x^2+1296*_C1^2+96)^(1/2))^(1
/3)+1/4*x^2+1/2*I*3^(1/2)*(1/12*(-108*x^2-144*x^3-432*_C1+27*x^6+12*(-54*x^9-162
*_C1*x^6+144*x^6+216*x^5+864*_C1*x^3-27*x^4+648*_C1*x^2+1296*_C1^2+96)^(1/2))^(1
/3)+12*(1/6-1/16*x^4)/(-108*x^2-144*x^3-432*_C1+27*x^6+12*(-54*x^9-162*_C1*x^6+1
44*x^6+216*x^5+864*_C1*x^3-27*x^4+648*_C1*x^2+1296*_C1^2+96)^(1/2))^(1/3))]