##### 4.13.29 $$\left (x^2-3 y(x)^2\right ) y'(x)+2 x y(x)+1=0$$

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

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.362842 (sec), leaf count = 307

$\left \{\left \{y(x)\to -\frac {\sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{6 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{6 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}\right \}\right \}$

Maple
cpu = 0.033 (sec), leaf count = 392

$\left [y \left (x \right ) = \frac {\left (108 x +108 \textit {\_C1} +12 \sqrt {-12 x^{6}+81 \textit {\_C1}^{2}+162 x \textit {\_C1} +81 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (108 x +108 \textit {\_C1} +12 \sqrt {-12 x^{6}+81 \textit {\_C1}^{2}+162 x \textit {\_C1} +81 x^{2}}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (108 x +108 \textit {\_C1} +12 \sqrt {-12 x^{6}+81 \textit {\_C1}^{2}+162 x \textit {\_C1} +81 x^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{\left (108 x +108 \textit {\_C1} +12 \sqrt {-12 x^{6}+81 \textit {\_C1}^{2}+162 x \textit {\_C1} +81 x^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (108 x +108 \textit {\_C1} +12 \sqrt {-12 x^{6}+81 \textit {\_C1}^{2}+162 x \textit {\_C1} +81 x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{\left (108 x +108 \textit {\_C1} +12 \sqrt {-12 x^{6}+81 \textit {\_C1}^{2}+162 x \textit {\_C1} +81 x^{2}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (108 x +108 \textit {\_C1} +12 \sqrt {-12 x^{6}+81 \textit {\_C1}^{2}+162 x \textit {\_C1} +81 x^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{\left (108 x +108 \textit {\_C1} +12 \sqrt {-12 x^{6}+81 \textit {\_C1}^{2}+162 x \textit {\_C1} +81 x^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (108 x +108 \textit {\_C1} +12 \sqrt {-12 x^{6}+81 \textit {\_C1}^{2}+162 x \textit {\_C1} +81 x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{\left (108 x +108 \textit {\_C1} +12 \sqrt {-12 x^{6}+81 \textit {\_C1}^{2}+162 x \textit {\_C1} +81 x^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -((2^(1/3)*x^2)/(-27*x + Sqrt[-108*x^6 + 729*(x - C[1])^2] + 27*C[1])^
(1/3)) - (-27*x + Sqrt[-108*x^6 + 729*(x - C[1])^2] + 27*C[1])^(1/3)/(3*2^(1/3))
}, {y[x] -> ((1 + I*Sqrt[3])*x^2)/(2^(2/3)*(-27*x + Sqrt[-108*x^6 + 729*(x - C[1
])^2] + 27*C[1])^(1/3)) + ((1 - I*Sqrt[3])*(-27*x + Sqrt[-108*x^6 + 729*(x - C[1
])^2] + 27*C[1])^(1/3))/(6*2^(1/3))}, {y[x] -> ((1 - I*Sqrt[3])*x^2)/(2^(2/3)*(-
27*x + Sqrt[-108*x^6 + 729*(x - C[1])^2] + 27*C[1])^(1/3)) + ((1 + I*Sqrt[3])*(-
27*x + Sqrt[-108*x^6 + 729*(x - C[1])^2] + 27*C[1])^(1/3))/(6*2^(1/3))}}

Maple raw input

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

Maple raw output

[y(x) = 1/6*(108*x+108*_C1+12*(-12*x^6+81*_C1^2+162*_C1*x+81*x^2)^(1/2))^(1/3)+2
*x^2/(108*x+108*_C1+12*(-12*x^6+81*_C1^2+162*_C1*x+81*x^2)^(1/2))^(1/3), y(x) =
-1/12*(108*x+108*_C1+12*(-12*x^6+81*_C1^2+162*_C1*x+81*x^2)^(1/2))^(1/3)-x^2/(10
8*x+108*_C1+12*(-12*x^6+81*_C1^2+162*_C1*x+81*x^2)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1
/6*(108*x+108*_C1+12*(-12*x^6+81*_C1^2+162*_C1*x+81*x^2)^(1/2))^(1/3)-2*x^2/(108
*x+108*_C1+12*(-12*x^6+81*_C1^2+162*_C1*x+81*x^2)^(1/2))^(1/3)), y(x) = -1/12*(1
08*x+108*_C1+12*(-12*x^6+81*_C1^2+162*_C1*x+81*x^2)^(1/2))^(1/3)-x^2/(108*x+108*
_C1+12*(-12*x^6+81*_C1^2+162*_C1*x+81*x^2)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/6*(108*
x+108*_C1+12*(-12*x^6+81*_C1^2+162*_C1*x+81*x^2)^(1/2))^(1/3)-2*x^2/(108*x+108*_
C1+12*(-12*x^6+81*_C1^2+162*_C1*x+81*x^2)^(1/2))^(1/3))]