ODE
\[ 8 x^3 y'(x)^3+12 x^2 y(x) y'(x)^2-\left (1-6 x y(x)^2\right ) y'(x)+y(x)^3=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _dAlembert]
Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)
Mathematica ✓
cpu = 537.291 (sec), leaf count = 1874
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {24 K[1]^6+9 \left (1+i \sqrt {3}\right ) y(x) \sqrt [3]{6 \sqrt {3} \sqrt {K[1]^9 \left (27 K[1] y(x)^2-2\right )}-54 K[1]^5 y(x)} K[1]^5-\sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {K[1]^9 \left (27 K[1] y(x)^2-2\right )}-18 K[1]^5 y(x)\right )^{2/3} K[1]^3+\left (3 i+\sqrt {3}\right ) \sqrt {K[1]^9 \left (27 K[1] y(x)^2-2\right )} \sqrt [3]{6 \sqrt {3} \sqrt {K[1]^9 \left (27 K[1] y(x)^2-2\right )}-54 K[1]^5 y(x)}}{36 K[1]^7}dK[1]+\int _1^{y(x)}\frac {-18 K[2] \int _1^x\frac {-2 \sqrt [3]{6} \left (3 i+\sqrt {3}\right ) K[1]^7+9 i 2^{2/3} \sqrt [6]{3} \left (i+\sqrt {3}\right ) K[2] \sqrt [3]{\sqrt {3} \sqrt {K[1]^9 \left (27 K[1] K[2]^2-2\right )}-9 K[1]^5 K[2]} K[1]^6+2^{2/3} \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) \sqrt {K[1]^9 \left (27 K[1] K[2]^2-2\right )} \sqrt [3]{\sqrt {3} \sqrt {K[1]^9 \left (27 K[1] K[2]^2-2\right )}-9 K[1]^5 K[2]} K[1]}{6 \sqrt {K[1]^9 \left (27 K[1] K[2]^2-2\right )} \left (\sqrt {3} \sqrt {K[1]^9 \left (27 K[1] K[2]^2-2\right )}-9 K[1]^5 K[2]\right )^{2/3}}dK[1] x^6-12 x^6+9 \left (1+i \sqrt {3}\right ) K[2] \sqrt [3]{6 \sqrt {3} \sqrt {x^9 \left (27 x K[2]^2-2\right )}-54 x^5 K[2]} x^5-\sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {x^9 \left (27 x K[2]^2-2\right )}-18 x^5 K[2]\right )^{2/3} x^3+\left (3 i+\sqrt {3}\right ) \sqrt {x^9 \left (27 x K[2]^2-2\right )} \sqrt [3]{6 \sqrt {3} \sqrt {x^9 \left (27 x K[2]^2-2\right )}-54 x^5 K[2]}}{18 x^6 K[2]}dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {24 K[3]^6+9 \left (1-i \sqrt {3}\right ) y(x) \sqrt [3]{6 \sqrt {3} \sqrt {K[3]^9 \left (27 K[3] y(x)^2-2\right )}-54 K[3]^5 y(x)} K[3]^5-\sqrt [6]{3} \left (3 i+\sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {K[3]^9 \left (27 K[3] y(x)^2-2\right )}-18 K[3]^5 y(x)\right )^{2/3} K[3]^3+\left (-3 i+\sqrt {3}\right ) \sqrt {K[3]^9 \left (27 K[3] y(x)^2-2\right )} \sqrt [3]{6 \sqrt {3} \sqrt {K[3]^9 \left (27 K[3] y(x)^2-2\right )}-54 K[3]^5 y(x)}}{36 K[3]^7}dK[3]+\int _1^{y(x)}\frac {-18 K[4] \int _1^x\frac {-2 \sqrt [3]{6} \left (-3 i+\sqrt {3}\right ) K[3]^7-9 i 2^{2/3} \sqrt [6]{3} \left (-i+\sqrt {3}\right ) K[4] \sqrt [3]{\sqrt {3} \sqrt {K[3]^9 \left (27 K[3] K[4]^2-2\right )}-9 K[3]^5 K[4]} K[3]^6+2^{2/3} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) \sqrt {K[3]^9 \left (27 K[3] K[4]^2-2\right )} \sqrt [3]{\sqrt {3} \sqrt {K[3]^9 \left (27 K[3] K[4]^2-2\right )}-9 K[3]^5 K[4]} K[3]}{6 \sqrt {K[3]^9 \left (27 K[3] K[4]^2-2\right )} \left (\sqrt {3} \sqrt {K[3]^9 \left (27 K[3] K[4]^2-2\right )}-9 K[3]^5 K[4]\right )^{2/3}}dK[3] x^6-12 x^6+9 \left (1-i \sqrt {3}\right ) K[4] \sqrt [3]{6 \sqrt {3} \sqrt {x^9 \left (27 x K[4]^2-2\right )}-54 x^5 K[4]} x^5-\sqrt [6]{3} \left (3 i+\sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {x^9 \left (27 x K[4]^2-2\right )}-18 x^5 K[4]\right )^{2/3} x^3+\left (-3 i+\sqrt {3}\right ) \sqrt {x^9 \left (27 x K[4]^2-2\right )} \sqrt [3]{6 \sqrt {3} \sqrt {x^9 \left (27 x K[4]^2-2\right )}-54 x^5 K[4]}}{18 x^6 K[4]}dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {12 K[5]^6-9 y(x) \sqrt [3]{6 \sqrt {3} \sqrt {K[5]^9 \left (27 K[5] y(x)^2-2\right )}-54 K[5]^5 y(x)} K[5]^5+\left (6 \sqrt {3} \sqrt {K[5]^9 \left (27 K[5] y(x)^2-2\right )}-54 K[5]^5 y(x)\right )^{2/3} K[5]^3-3^{5/6} \sqrt {K[5]^9 \left (27 K[5] y(x)^2-2\right )} \sqrt [3]{2 \sqrt {3} \sqrt {K[5]^9 \left (27 K[5] y(x)^2-2\right )}-18 K[5]^5 y(x)}}{18 K[5]^7}dK[5]+\int _1^{y(x)}-\frac {9 K[6] \int _1^x\frac {2 \sqrt [3]{2} 3^{5/6} K[5]^7+9\ 2^{2/3} \sqrt [6]{3} K[6] \sqrt [3]{\sqrt {3} \sqrt {K[5]^9 \left (27 K[5] K[6]^2-2\right )}-9 K[5]^5 K[6]} K[5]^6-6^{2/3} \sqrt {K[5]^9 \left (27 K[5] K[6]^2-2\right )} \sqrt [3]{\sqrt {3} \sqrt {K[5]^9 \left (27 K[5] K[6]^2-2\right )}-9 K[5]^5 K[6]} K[5]}{3 \sqrt {K[5]^9 \left (27 K[5] K[6]^2-2\right )} \left (\sqrt {3} \sqrt {K[5]^9 \left (27 K[5] K[6]^2-2\right )}-9 K[5]^5 K[6]\right )^{2/3}}dK[5] x^6+6 x^6+9 K[6] \sqrt [3]{6 \sqrt {3} \sqrt {x^9 \left (27 x K[6]^2-2\right )}-54 x^5 K[6]} x^5-\left (6 \sqrt {3} \sqrt {x^9 \left (27 x K[6]^2-2\right )}-54 x^5 K[6]\right )^{2/3} x^3+3^{5/6} \sqrt {x^9 \left (27 x K[6]^2-2\right )} \sqrt [3]{2 \sqrt {3} \sqrt {x^9 \left (27 x K[6]^2-2\right )}-18 x^5 K[6]}}{9 x^6 K[6]}dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 2.976 (sec), leaf count = 3883
\[\text {Expression too large to display}\] Mathematica raw input
DSolve[y[x]^3 - (1 - 6*x*y[x]^2)*y'[x] + 12*x^2*y[x]*y'[x]^2 + 8*x^3*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][(24*K[1]^6 - 3^(1/6)*(-3*I + Sqrt[3])*K[1]^3*(-18*K[1]^5*y[x] + 2*Sqrt[3]*Sqrt[K[1]^9*(-2 + 27*K[1]*y[x]^2)])^(2/3) + 9*(1 + I*Sqrt[3])*K[1]^5*y[x]*(-54*K[1]^5*y[x] + 6*Sqrt[3]*Sqrt[K[1]^9*(-2 + 27*K[1]*y[x]^2)])^(1/3) + (3*I + Sqrt[3])*Sqrt[K[1]^9*(-2 + 27*K[1]*y[x]^2)]*(-54*K[1]^5*y[x] + 6*Sqrt[3]*Sqrt[K[1]^9*(-2 + 27*K[1]*y[x]^2)])^(1/3))/(36*K[1]^7), {K[1], 1, x}] + Inactive[Integrate][(-12*x^6 - 3^(1/6)*(-3*I + Sqrt[3])*x^3*(-18*x^5*K[2] + 2*Sqrt[3]*Sqrt[x^9*(-2 + 27*x*K[2]^2)])^(2/3) + 9*(1 + I*Sqrt[3])*x^5*K[2]*(-54*x^5*K[2] + 6*Sqrt[3]*Sqrt[x^9*(-2 + 27*x*K[2]^2)])^(1/3) + (3*I + Sqrt[3])*Sqrt[x^9*(-2 + 27*x*K[2]^2)]*(-54*x^5*K[2] + 6*Sqrt[3]*Sqrt[x^9*(-2 + 27*x*K[2]^2)])^(1/3) - 18*x^6*K[2]*Inactive[Integrate][(-2*6^(1/3)*(3*I + Sqrt[3])*K[1]^7 + (9*I)*2^(2/3)*3^(1/6)*(I + Sqrt[3])*K[1]^6*K[2]*(-9*K[1]^5*K[2] + Sqrt[3]*Sqrt[K[1]^9*(-2 + 27*K[1]*K[2]^2)])^(1/3) + 2^(2/3)*3^(1/6)*(-3*I + Sqrt[3])*K[1]*Sqrt[K[1]^9*(-2 + 27*K[1]*K[2]^2)]*(-9*K[1]^5*K[2] + Sqrt[3]*Sqrt[K[1]^9*(-2 + 27*K[1]*K[2]^2)])^(1/3))/(6*Sqrt[K[1]^9*(-2 + 27*K[1]*K[2]^2)]*(-9*K[1]^5*K[2] + Sqrt[3]*Sqrt[K[1]^9*(-2 + 27*K[1]*K[2]^2)])^(2/3)), {K[1], 1, x}])/(18*x^6*K[2]), {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(24*K[3]^6 - 3^(1/6)*(3*I + Sqrt[3])*K[3]^3*(-18*K[3]^5*y[x] + 2*Sqrt[3]*Sqrt[K[3]^9*(-2 + 27*K[3]*y[x]^2)])^(2/3) + 9*(1 - I*Sqrt[3])*K[3]^5*y[x]*(-54*K[3]^5*y[x] + 6*Sqrt[3]*Sqrt[K[3]^9*(-2 + 27*K[3]*y[x]^2)])^(1/3) + (-3*I + Sqrt[3])*Sqrt[K[3]^9*(-2 + 27*K[3]*y[x]^2)]*(-54*K[3]^5*y[x] + 6*Sqrt[3]*Sqrt[K[3]^9*(-2 + 27*K[3]*y[x]^2)])^(1/3))/(36*K[3]^7), {K[3], 1, x}] + Inactive[Integrate][(-12*x^6 - 3^(1/6)*(3*I + Sqrt[3])*x^3*(-18*x^5*K[4] + 2*Sqrt[3]*Sqrt[x^9*(-2 + 27*x*K[4]^2)])^(2/3) + 9*(1 - I*Sqrt[3])*x^5*K[4]*(-54*x^5*K[4] + 6*Sqrt[3]*Sqrt[x^9*(-2 + 27*x*K[4]^2)])^(1/3) + (-3*I + Sqrt[3])*Sqrt[x^9*(-2 + 27*x*K[4]^2)]*(-54*x^5*K[4] + 6*Sqrt[3]*Sqrt[x^9*(-2 + 27*x*K[4]^2)])^(1/3) - 18*x^6*K[4]*Inactive[Integrate][(-2*6^(1/3)*(-3*I + Sqrt[3])*K[3]^7 - (9*I)*2^(2/3)*3^(1/6)*(-I + Sqrt[3])*K[3]^6*K[4]*(-9*K[3]^5*K[4] + Sqrt[3]*Sqrt[K[3]^9*(-2 + 27*K[3]*K[4]^2)])^(1/3) + 2^(2/3)*3^(1/6)*(3*I + Sqrt[3])*K[3]*Sqrt[K[3]^9*(-2 + 27*K[3]*K[4]^2)]*(-9*K[3]^5*K[4] + Sqrt[3]*Sqrt[K[3]^9*(-2 + 27*K[3]*K[4]^2)])^(1/3))/(6*Sqrt[K[3]^9*(-2 + 27*K[3]*K[4]^2)]*(-9*K[3]^5*K[4] + Sqrt[3]*Sqrt[K[3]^9*(-2 + 27*K[3]*K[4]^2)])^(2/3)), {K[3], 1, x}])/(18*x^6*K[4]), {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(12*K[5]^6 - 3^(5/6)*Sqrt[K[5]^9*(-2 + 27*K[5]*y[x]^2)]*(-18*K[5]^5*y[x] + 2*Sqrt[3]*Sqrt[K[5]^9*(-2 + 27*K[5]*y[x]^2)])^(1/3) - 9*K[5]^5*y[x]*(-54*K[5]^5*y[x] + 6*Sqrt[3]*Sqrt[K[5]^9*(-2 + 27*K[5]*y[x]^2)])^(1/3) + K[5]^3*(-54*K[5]^5*y[x] + 6*Sqrt[3]*Sqrt[K[5]^9*(-2 + 27*K[5]*y[x]^2)])^(2/3))/(18*K[5]^7), {K[5], 1, x}] + Inactive[Integrate][-1/9*(6*x^6 + 3^(5/6)*Sqrt[x^9*(-2 + 27*x*K[6]^2)]*(-18*x^5*K[6] + 2*Sqrt[3]*Sqrt[x^9*(-2 + 27*x*K[6]^2)])^(1/3) + 9*x^5*K[6]*(-54*x^5*K[6] + 6*Sqrt[3]*Sqrt[x^9*(-2 + 27*x*K[6]^2)])^(1/3) - x^3*(-54*x^5*K[6] + 6*Sqrt[3]*Sqrt[x^9*(-2 + 27*x*K[6]^2)])^(2/3) + 9*x^6*K[6]*Inactive[Integrate][(2*2^(1/3)*3^(5/6)*K[5]^7 + 9*2^(2/3)*3^(1/6)*K[5]^6*K[6]*(-9*K[5]^5*K[6] + Sqrt[3]*Sqrt[K[5]^9*(-2 + 27*K[5]*K[6]^2)])^(1/3) - 6^(2/3)*K[5]*Sqrt[K[5]^9*(-2 + 27*K[5]*K[6]^2)]*(-9*K[5]^5*K[6] + Sqrt[3]*Sqrt[K[5]^9*(-2 + 27*K[5]*K[6]^2)])^(1/3))/(3*Sqrt[K[5]^9*(-2 + 27*K[5]*K[6]^2)]*(-9*K[5]^5*K[6] + Sqrt[3]*Sqrt[K[5]^9*(-2 + 27*K[5]*K[6]^2)])^(2/3)), {K[5], 1, x}])/(x^6*K[6]), {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(8*x^3*diff(y(x),x)^3+12*x^2*y(x)*diff(y(x),x)^2-(1-6*x*y(x)^2)*diff(y(x),x)+y(x)^3 = 0, y(x))
Maple raw output
[x-1/9*(ln(-1/24*(I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-I*6^(2/3)*3^(1/2)*x+6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))+9*_C1)*24^(2/3)/((I*6^(2/3)*3^(1/2)*x-I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3))/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))^(2/3) = 0, -_C1/((6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-6*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))^(2/3)+x-1/9*ln(1/12*(6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-6*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))*12^(2/3)/((-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+6*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x)^2/x^4/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(2/3))^(1/3) = 0, -_C1/((I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-I*6^(2/3)*3^(1/2)*x-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))^(2/3)+x-4/9*ln(1/24*(I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-I*6^(2/3)*3^(1/2)*x-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))*3^(2/3)/((-I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+I*6^(2/3)*3^(1/2)*x+6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)^2/x^4/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(2/3))^(1/3) = 0, x-24^(2/3)/((I*6^(2/3)*3^(1/2)*x-I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3))/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))^(2/3)*(Intat(-1/18*(1+I*3^(1/2))/_a,_a = -1/24*(I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-I*6^(2/3)*3^(1/2)*x+6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))+_C1) = 0, -_C1/((6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-6*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))^(2/3)+x-Intat(-1/18*(1+I*3^(1/2))/_a,_a = 1/12*(6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-6*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))*12^(2/3)/((-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+6*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x)^2/x^4/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(2/3))^(1/3) = 0, -_C1/((I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-I*6^(2/3)*3^(1/2)*x-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))^(2/3)+x-4*Intat(-1/18*(1+I*3^(1/2))/_a,_a = 1/24*(I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-I*6^(2/3)*3^(1/2)*x-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))*3^(2/3)/((-I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+I*6^(2/3)*3^(1/2)*x+6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)^2/x^4/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(2/3))^(1/3) = 0, -_C1/((6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-6*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))^(2/3)+x-Intat(1/18*(-1+I*3^(1/2))/_a,_a = 1/12*(6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-6*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))*12^(2/3)/((-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+6*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x)^2/x^4/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(2/3))^(1/3) = 0, -_C1/((I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-I*6^(2/3)*3^(1/2)*x-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))^(2/3)+x-4*Intat(1/18*(-1+I*3^(1/2))/_a,_a = 1/24*(I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-I*6^(2/3)*3^(1/2)*x-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))*3^(2/3)/((-I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+I*6^(2/3)*3^(1/2)*x+6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)^2/x^4/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(2/3))^(1/3) = 0, -_C1*6^(2/3)/((I*6^(2/3)*3^(1/2)*x-I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3))/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))^(2/3)+x-4*Intat(1/18*(-1+I*3^(1/2))/_a,_a = -1/24*(I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-I*6^(2/3)*3^(1/2)*x+6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)+12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)+6^(2/3)*x)/x^2/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3))*3^(2/3)/((I*6^(2/3)*3^(1/2)*x-I*3^(1/2)*6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3)-12*y(x)*x*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(1/3)-6^(2/3)*x-6^(1/3)*((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))^2*x^4)^(1/3))^2/x^4/((3^(1/2)*((27*x*y(x)^2-2)/x)^(1/2)-9*y(x))*x^2)^(2/3))^(1/3) = 0]