ODE
\[ 2 y(x) y'(x)^3-3 x y'(x)+2 y(x)=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 69.1999 (sec), leaf count = 2177
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {-2 \left (2 \sqrt {4 y(x)^6-2 K[1]^3 y(x)^3}-4 y(x)^3\right )^{2/3} y(x)^4+\sqrt [6]{2} \left (\sqrt {2} K[1]^3-2 \sqrt {2 y(x)^6-K[1]^3 y(x)^3}\right ) \left (\sqrt {4 y(x)^6-2 K[1]^3 y(x)^3}-2 y(x)^3\right )^{2/3} y(x)-2^{5/6} K[1]^2 \sqrt {2 y(x)^6-K[1]^3 y(x)^3} \sqrt [3]{\sqrt {4 y(x)^6-2 K[1]^3 y(x)^3}-2 y(x)^3}}{2 K[1]^2 y(x)^2 \left (K[1]^3-2 y(x)^3\right )}dK[1]+\int _1^{y(x)}\frac {4 x K[2]^5+2 x \left (x^3-2 K[2]^3\right ) \int _1^x-\frac {4\ 2^{5/6} K[2]^6+2^{5/6} K[1]^3 K[2]^3-4 \sqrt [3]{2} \sqrt {2 K[2]^6-K[1]^3 K[2]^3} K[2]^3+\sqrt [6]{2} K[1] \left (6 K[2]^3-\sqrt {4 K[2]^6-2 K[1]^3 K[2]^3}\right ) \sqrt [3]{\sqrt {4 K[2]^6-2 K[1]^3 K[2]^3}-2 K[2]^3} K[2]}{2 \left (K[1]^3-2 K[2]^3\right ) \sqrt {2 K[2]^6-K[1]^3 K[2]^3} \left (\sqrt {4 K[2]^6-2 K[1]^3 K[2]^3}-2 K[2]^3\right )^{2/3}}dK[1] K[2]^3-2 x^4 K[2]^2-2 \sqrt [6]{2} \left (\sqrt {2} K[2]^3+\sqrt {2 K[2]^6-x^3 K[2]^3}\right ) \left (\sqrt {4 K[2]^6-2 x^3 K[2]^3}-2 K[2]^3\right )^{2/3} K[2]+x^3 \left (2 \sqrt {4 K[2]^6-2 x^3 K[2]^3}-4 K[2]^3\right )^{2/3} K[2]-2^{5/6} x^2 \sqrt {2 K[2]^6-x^3 K[2]^3} \sqrt [3]{\sqrt {4 K[2]^6-2 x^3 K[2]^3}-2 K[2]^3}}{4 x K[2]^6-2 x^4 K[2]^3}dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {2\ 2^{2/3} \left (1+i \sqrt {3}\right ) \left (\sqrt {4 y(x)^6-2 K[3]^3 y(x)^3}-2 y(x)^3\right )^{2/3} y(x)^4+\sqrt [6]{2} \left (1+i \sqrt {3}\right ) \left (2 \sqrt {2 y(x)^6-K[3]^3 y(x)^3}-\sqrt {2} K[3]^3\right ) \left (\sqrt {4 y(x)^6-2 K[3]^3 y(x)^3}-2 y(x)^3\right )^{2/3} y(x)+2^{5/6} \left (1-i \sqrt {3}\right ) K[3]^2 \sqrt {2 y(x)^6-K[3]^3 y(x)^3} \sqrt [3]{\sqrt {4 y(x)^6-2 K[3]^3 y(x)^3}-2 y(x)^3}}{4 K[3]^2 y(x)^2 \left (K[3]^3-2 y(x)^3\right )}dK[3]+\int _1^{y(x)}\frac {8 x K[4]^5+4 x \left (x^3-2 K[4]^3\right ) \int _1^x-\frac {i 2^{5/6} \left (i+\sqrt {3}\right ) K[3]^3 K[4]^3+4 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) \left (\sqrt {2 K[4]^6-K[3]^3 K[4]^3}-\sqrt {2} K[4]^3\right ) K[4]^3+\sqrt [6]{2} \left (1+i \sqrt {3}\right ) K[3] \left (\sqrt {4 K[4]^6-2 K[3]^3 K[4]^3}-6 K[4]^3\right ) \sqrt [3]{\sqrt {4 K[4]^6-2 K[3]^3 K[4]^3}-2 K[4]^3} K[4]}{4 \left (K[3]^3-2 K[4]^3\right ) \sqrt {2 K[4]^6-K[3]^3 K[4]^3} \left (\sqrt {4 K[4]^6-2 K[3]^3 K[4]^3}-2 K[4]^3\right )^{2/3}}dK[3] K[4]^3-4 x^4 K[4]^2+2 \sqrt [6]{2} \left (1+i \sqrt {3}\right ) \left (\sqrt {2} K[4]^3+\sqrt {2 K[4]^6-x^3 K[4]^3}\right ) \left (\sqrt {4 K[4]^6-2 x^3 K[4]^3}-2 K[4]^3\right )^{2/3} K[4]+\left (-1-i \sqrt {3}\right ) x^3 \left (2 \sqrt {4 K[4]^6-2 x^3 K[4]^3}-4 K[4]^3\right )^{2/3} K[4]+2^{5/6} \left (1-i \sqrt {3}\right ) x^2 \sqrt {2 K[4]^6-x^3 K[4]^3} \sqrt [3]{\sqrt {4 K[4]^6-2 x^3 K[4]^3}-2 K[4]^3}}{8 x K[4]^6-4 x^4 K[4]^3}dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {2\ 2^{2/3} \left (1-i \sqrt {3}\right ) \left (\sqrt {4 y(x)^6-2 K[5]^3 y(x)^3}-2 y(x)^3\right )^{2/3} y(x)^4+\sqrt [6]{2} \left (1-i \sqrt {3}\right ) \left (2 \sqrt {2 y(x)^6-K[5]^3 y(x)^3}-\sqrt {2} K[5]^3\right ) \left (\sqrt {4 y(x)^6-2 K[5]^3 y(x)^3}-2 y(x)^3\right )^{2/3} y(x)+2^{5/6} \left (1+i \sqrt {3}\right ) K[5]^2 \sqrt {2 y(x)^6-K[5]^3 y(x)^3} \sqrt [3]{\sqrt {4 y(x)^6-2 K[5]^3 y(x)^3}-2 y(x)^3}}{4 K[5]^2 y(x)^2 \left (K[5]^3-2 y(x)^3\right )}dK[5]+\int _1^{y(x)}\frac {8 x K[6]^5+4 x \left (x^3-2 K[6]^3\right ) \int _1^x-\frac {2^{5/6} \left (-1-i \sqrt {3}\right ) K[5]^3 K[6]^3+4 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (\sqrt {2 K[6]^6-K[5]^3 K[6]^3}-\sqrt {2} K[6]^3\right ) K[6]^3+\sqrt [6]{2} \left (1-i \sqrt {3}\right ) K[5] \left (\sqrt {4 K[6]^6-2 K[5]^3 K[6]^3}-6 K[6]^3\right ) \sqrt [3]{\sqrt {4 K[6]^6-2 K[5]^3 K[6]^3}-2 K[6]^3} K[6]}{4 \left (K[5]^3-2 K[6]^3\right ) \sqrt {2 K[6]^6-K[5]^3 K[6]^3} \left (\sqrt {4 K[6]^6-2 K[5]^3 K[6]^3}-2 K[6]^3\right )^{2/3}}dK[5] K[6]^3-4 x^4 K[6]^2+2 \sqrt [6]{2} \left (1-i \sqrt {3}\right ) \left (\sqrt {2} K[6]^3+\sqrt {2 K[6]^6-x^3 K[6]^3}\right ) \left (\sqrt {4 K[6]^6-2 x^3 K[6]^3}-2 K[6]^3\right )^{2/3} K[6]+i \left (i+\sqrt {3}\right ) x^3 \left (2 \sqrt {4 K[6]^6-2 x^3 K[6]^3}-4 K[6]^3\right )^{2/3} K[6]+2^{5/6} \left (1+i \sqrt {3}\right ) x^2 \sqrt {2 K[6]^6-x^3 K[6]^3} \sqrt [3]{\sqrt {4 K[6]^6-2 x^3 K[6]^3}-2 K[6]^3}}{8 x K[6]^6-4 x^4 K[6]^3}dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.202 (sec), leaf count = 731
\[\left [y \left (x \right ) = \frac {2^{\frac {2}{3}} x}{2}, y \left (x \right ) = \left (-\frac {2^{\frac {2}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{4}\right ) x, y \left (x \right ) = \left (-\frac {2^{\frac {2}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{4}\right ) x, y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-2 \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}} \textit {\_a}^{3}-2 \textit {\_a}^{3}+\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {2}{3}}+\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}+1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right ) \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} +\textit {\_C1} \right ) x, y \left (x \right ) = \RootOf \left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {2 i \sqrt {3}\, \textit {\_a}^{3}+i \sqrt {3}\, \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {2}{3}}-4 \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}} \textit {\_a}^{3}+2 \textit {\_a}^{3}-i \sqrt {3}-\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {2}{3}}+2 \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}-1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right ) \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} +2 \textit {\_C1} \right ) x, y \left (x \right ) = \RootOf \left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 i \sqrt {3}\, \textit {\_a}^{3}+i \sqrt {3}\, \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {2}{3}}+4 \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}} \textit {\_a}^{3}-2 \textit {\_a}^{3}-i \sqrt {3}+\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {2}{3}}-2 \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}+1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right ) \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} \right )+2 \textit {\_C1} \right ) x\right ]\] Mathematica raw input
DSolve[2*y[x] - 3*x*y'[x] + 2*y[x]*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][(-(2^(5/6)*K[1]^2*Sqrt[-(K[1]^3*y[x]^3) + 2*y[x]^6]*(-2*y[x]^3 + Sqrt[-2*K[1]^3*y[x]^3 + 4*y[x]^6])^(1/3)) + 2^(1/6)*y[x]*(Sqrt[2]*K[1]^3 - 2*Sqrt[-(K[1]^3*y[x]^3) + 2*y[x]^6])*(-2*y[x]^3 + Sqrt[-2*K[1]^3*y[x]^3 + 4*y[x]^6])^(2/3) - 2*y[x]^4*(-4*y[x]^3 + 2*Sqrt[-2*K[1]^3*y[x]^3 + 4*y[x]^6])^(2/3))/(2*K[1]^2*y[x]^2*(K[1]^3 - 2*y[x]^3)), {K[1], 1, x}] + Inactive[Integrate][(-2*x^4*K[2]^2 + 4*x*K[2]^5 - 2^(5/6)*x^2*Sqrt[-(x^3*K[2]^3) + 2*K[2]^6]*(-2*K[2]^3 + Sqrt[-2*x^3*K[2]^3 + 4*K[2]^6])^(1/3) - 2*2^(1/6)*K[2]*(Sqrt[2]*K[2]^3 + Sqrt[-(x^3*K[2]^3) + 2*K[2]^6])*(-2*K[2]^3 + Sqrt[-2*x^3*K[2]^3 + 4*K[2]^6])^(2/3) + x^3*K[2]*(-4*K[2]^3 + 2*Sqrt[-2*x^3*K[2]^3 + 4*K[2]^6])^(2/3) + 2*x*K[2]^3*(x^3 - 2*K[2]^3)*Inactive[Integrate][-1/2*(2^(5/6)*K[1]^3*K[2]^3 + 4*2^(5/6)*K[2]^6 - 4*2^(1/3)*K[2]^3*Sqrt[-(K[1]^3*K[2]^3) + 2*K[2]^6] + 2^(1/6)*K[1]*K[2]*(6*K[2]^3 - Sqrt[-2*K[1]^3*K[2]^3 + 4*K[2]^6])*(-2*K[2]^3 + Sqrt[-2*K[1]^3*K[2]^3 + 4*K[2]^6])^(1/3))/((K[1]^3 - 2*K[2]^3)*Sqrt[-(K[1]^3*K[2]^3) + 2*K[2]^6]*(-2*K[2]^3 + Sqrt[-2*K[1]^3*K[2]^3 + 4*K[2]^6])^(2/3)), {K[1], 1, x}])/(-2*x^4*K[2]^3 + 4*x*K[2]^6), {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(2^(5/6)*(1 - I*Sqrt[3])*K[3]^2*Sqrt[-(K[3]^3*y[x]^3) + 2*y[x]^6]*(-2*y[x]^3 + Sqrt[-2*K[3]^3*y[x]^3 + 4*y[x]^6])^(1/3) + 2*2^(2/3)*(1 + I*Sqrt[3])*y[x]^4*(-2*y[x]^3 + Sqrt[-2*K[3]^3*y[x]^3 + 4*y[x]^6])^(2/3) + 2^(1/6)*(1 + I*Sqrt[3])*y[x]*(-(Sqrt[2]*K[3]^3) + 2*Sqrt[-(K[3]^3*y[x]^3) + 2*y[x]^6])*(-2*y[x]^3 + Sqrt[-2*K[3]^3*y[x]^3 + 4*y[x]^6])^(2/3))/(4*K[3]^2*y[x]^2*(K[3]^3 - 2*y[x]^3)), {K[3], 1, x}] + Inactive[Integrate][(-4*x^4*K[4]^2 + 8*x*K[4]^5 + 2^(5/6)*(1 - I*Sqrt[3])*x^2*Sqrt[-(x^3*K[4]^3) + 2*K[4]^6]*(-2*K[4]^3 + Sqrt[-2*x^3*K[4]^3 + 4*K[4]^6])^(1/3) + 2*2^(1/6)*(1 + I*Sqrt[3])*K[4]*(Sqrt[2]*K[4]^3 + Sqrt[-(x^3*K[4]^3) + 2*K[4]^6])*(-2*K[4]^3 + Sqrt[-2*x^3*K[4]^3 + 4*K[4]^6])^(2/3) + (-1 - I*Sqrt[3])*x^3*K[4]*(-4*K[4]^3 + 2*Sqrt[-2*x^3*K[4]^3 + 4*K[4]^6])^(2/3) + 4*x*K[4]^3*(x^3 - 2*K[4]^3)*Inactive[Integrate][-1/4*(I*2^(5/6)*(I + Sqrt[3])*K[3]^3*K[4]^3 + 4*2^(1/3)*(1 - I*Sqrt[3])*K[4]^3*(-(Sqrt[2]*K[4]^3) + Sqrt[-(K[3]^3*K[4]^3) + 2*K[4]^6]) + 2^(1/6)*(1 + I*Sqrt[3])*K[3]*K[4]*(-6*K[4]^3 + Sqrt[-2*K[3]^3*K[4]^3 + 4*K[4]^6])*(-2*K[4]^3 + Sqrt[-2*K[3]^3*K[4]^3 + 4*K[4]^6])^(1/3))/((K[3]^3 - 2*K[4]^3)*Sqrt[-(K[3]^3*K[4]^3) + 2*K[4]^6]*(-2*K[4]^3 + Sqrt[-2*K[3]^3*K[4]^3 + 4*K[4]^6])^(2/3)), {K[3], 1, x}])/(-4*x^4*K[4]^3 + 8*x*K[4]^6), {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(2^(5/6)*(1 + I*Sqrt[3])*K[5]^2*Sqrt[-(K[5]^3*y[x]^3) + 2*y[x]^6]*(-2*y[x]^3 + Sqrt[-2*K[5]^3*y[x]^3 + 4*y[x]^6])^(1/3) + 2*2^(2/3)*(1 - I*Sqrt[3])*y[x]^4*(-2*y[x]^3 + Sqrt[-2*K[5]^3*y[x]^3 + 4*y[x]^6])^(2/3) + 2^(1/6)*(1 - I*Sqrt[3])*y[x]*(-(Sqrt[2]*K[5]^3) + 2*Sqrt[-(K[5]^3*y[x]^3) + 2*y[x]^6])*(-2*y[x]^3 + Sqrt[-2*K[5]^3*y[x]^3 + 4*y[x]^6])^(2/3))/(4*K[5]^2*y[x]^2*(K[5]^3 - 2*y[x]^3)), {K[5], 1, x}] + Inactive[Integrate][(-4*x^4*K[6]^2 + 8*x*K[6]^5 + 2^(5/6)*(1 + I*Sqrt[3])*x^2*Sqrt[-(x^3*K[6]^3) + 2*K[6]^6]*(-2*K[6]^3 + Sqrt[-2*x^3*K[6]^3 + 4*K[6]^6])^(1/3) + 2*2^(1/6)*(1 - I*Sqrt[3])*K[6]*(Sqrt[2]*K[6]^3 + Sqrt[-(x^3*K[6]^3) + 2*K[6]^6])*(-2*K[6]^3 + Sqrt[-2*x^3*K[6]^3 + 4*K[6]^6])^(2/3) + I*(I + Sqrt[3])*x^3*K[6]*(-4*K[6]^3 + 2*Sqrt[-2*x^3*K[6]^3 + 4*K[6]^6])^(2/3) + 4*x*K[6]^3*(x^3 - 2*K[6]^3)*Inactive[Integrate][-1/4*(2^(5/6)*(-1 - I*Sqrt[3])*K[5]^3*K[6]^3 + 4*2^(1/3)*(1 + I*Sqrt[3])*K[6]^3*(-(Sqrt[2]*K[6]^3) + Sqrt[-(K[5]^3*K[6]^3) + 2*K[6]^6]) + 2^(1/6)*(1 - I*Sqrt[3])*K[5]*K[6]*(-6*K[6]^3 + Sqrt[-2*K[5]^3*K[6]^3 + 4*K[6]^6])*(-2*K[6]^3 + Sqrt[-2*K[5]^3*K[6]^3 + 4*K[6]^6])^(1/3))/((K[5]^3 - 2*K[6]^3)*Sqrt[-(K[5]^3*K[6]^3) + 2*K[6]^6]*(-2*K[6]^3 + Sqrt[-2*K[5]^3*K[6]^3 + 4*K[6]^6])^(2/3)), {K[5], 1, x}])/(-4*x^4*K[6]^3 + 8*x*K[6]^6), {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(2*y(x)*diff(y(x),x)^3-3*x*diff(y(x),x)+2*y(x) = 0, y(x))
Maple raw output
[y(x) = 1/2*2^(2/3)*x, y(x) = (-1/4*2^(2/3)-1/4*I*3^(1/2)*2^(2/3))*x, y(x) = (-1/4*2^(2/3)+1/4*I*3^(1/2)*2^(2/3))*x, y(x) = RootOf(-ln(x)+Intat((-2*((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(1/3)*_a^3-2*_a^3+((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(2/3)+((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(1/3)+1)/_a/(2*_a^3-1)/((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(1/3),_a = _Z)+_C1)*x, y(x) = RootOf(-2*ln(x)+Intat((2*I*3^(1/2)*_a^3+I*3^(1/2)*((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(2/3)-4*((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(1/3)*_a^3+2*_a^3-I*3^(1/2)-((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(2/3)+2*((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(1/3)-1)/_a/(2*_a^3-1)/((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(1/3),_a = _Z)+2*_C1)*x, y(x) = RootOf(-2*ln(x)-Intat((2*I*3^(1/2)*_a^3+I*3^(1/2)*((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(2/3)+4*((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(1/3)*_a^3-2*_a^3-I*3^(1/2)+((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(2/3)-2*((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(1/3)+1)/_a/(2*_a^3-1)/((2^(1/2)*(1/_a/(2*_a^3-1))^(1/2)*_a^2+1)*(2*_a^3-1)^2)^(1/3),_a = _Z)+2*_C1)*x]