ODE
\[ 8 y'(x)^3+12 y'(x)^2=27 (y(x)+x) \] ODE Classification
[[_homogeneous, `class C`], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 205.971 (sec), leaf count = 4555
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {1}{2} \left (\frac {3 \sqrt [3]{2} \sqrt {3} \sqrt {(K[1]+y(x)) (27 (K[1]+y(x))-4)} \left (-27 K[1]-27 y(x)+3 \sqrt {3} \sqrt {(K[1]+y(x)) (27 (K[1]+y(x))-4)}+2\right )^{2/3}}{27 K[1]+27 y(x)-4}+\sqrt [3]{2} \left (-27 K[1]-27 y(x)+3 \sqrt {3} \sqrt {(K[1]+y(x)) (27 (K[1]+y(x))-4)}+2\right )^{2/3}+\frac {3\ 2^{2/3} \sqrt {3} \sqrt {(K[1]+y(x)) (27 (K[1]+y(x))-4)} \sqrt [3]{-27 K[1]-27 y(x)+3 \sqrt {3} \sqrt {(K[1]+y(x)) (27 (K[1]+y(x))-4)}+2}}{27 K[1]+27 y(x)-4}+2^{2/3} \sqrt [3]{-27 K[1]-27 y(x)+3 \sqrt {3} \sqrt {(K[1]+y(x)) (27 (K[1]+y(x))-4)}+2}+2\right )dK[1]+\int _1^{y(x)}\left (\frac {\left (27 x+27 K[2]+3 \sqrt {3} \sqrt {27 x^2+(54 K[2]-4) x+K[2] (27 K[2]-4)}-4\right ) \sqrt [3]{-54 x-54 K[2]+6 \sqrt {3} \sqrt {27 x^2+(54 K[2]-4) x+K[2] (27 K[2]-4)}+4} \left (\sqrt [3]{-27 x-27 K[2]+3 \sqrt {3} \sqrt {27 x^2+(54 K[2]-4) x+K[2] (27 K[2]-4)}+2}+\sqrt [3]{2}\right )}{54 x+54 K[2]-8}-\int _1^x\frac {\sqrt [3]{2} \left (27 \sqrt {3} \sqrt [3]{-27 K[1]-27 K[2]+3 \sqrt {3} \sqrt {(K[1]+K[2]) (27 (K[1]+K[2])-4)}+2} K[1]+54 \sqrt [3]{2} \sqrt {3} K[1]+54 \sqrt [3]{2} \sqrt {3} K[2]-18 \sqrt [3]{2} \sqrt {(K[1]+K[2]) (27 (K[1]+K[2])-4)}+27 \sqrt {3} K[2] \sqrt [3]{-27 K[1]-27 K[2]+3 \sqrt {3} \sqrt {(K[1]+K[2]) (27 (K[1]+K[2])-4)}+2}-9 \sqrt {(K[1]+K[2]) (27 (K[1]+K[2])-4)} \sqrt [3]{-27 K[1]-27 K[2]+3 \sqrt {3} \sqrt {(K[1]+K[2]) (27 (K[1]+K[2])-4)}+2}+2 \sqrt {3} \sqrt [3]{-27 K[1]-27 K[2]+3 \sqrt {3} \sqrt {(K[1]+K[2]) (27 (K[1]+K[2])-4)}+2}-2 \sqrt [3]{2} \sqrt {3}\right )}{(27 K[1]+27 K[2]-4) \sqrt {(K[1]+K[2]) (27 (K[1]+K[2])-4)} \left (-27 K[1]-27 K[2]+3 \sqrt {3} \sqrt {(K[1]+K[2]) (27 (K[1]+K[2])-4)}+2\right )^{2/3}}dK[1]\right )dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {1}{4} \left (\frac {2 \sqrt [3]{2} \left (-27 K[3]-27 y(x)+3 \sqrt {3} \sqrt {27 y(x)^2+(54 K[3]-4) y(x)+K[3] (27 K[3]-4)}+2\right )^{2/3} \left (-27 \left (i+\sqrt {3}\right ) K[3]-27 \left (i+\sqrt {3}\right ) y(x)-3 i \sqrt {3} \sqrt {27 y(x)^2+(54 K[3]-4) y(x)+K[3] (27 K[3]-4)}-9 \sqrt {27 y(x)^2+(54 K[3]-4) y(x)+K[3] (27 K[3]-4)}+4 \sqrt {3}+4 i\right )}{\left (-i+\sqrt {3}\right ) (27 K[3]+27 y(x)-4)}-\frac {3\ 2^{2/3} \sqrt {3} \sqrt {27 y(x)^2+(54 K[3]-4) y(x)+K[3] (27 K[3]-4)} \sqrt [3]{-27 K[3]-27 y(x)+3 \sqrt {3} \sqrt {27 y(x)^2+(54 K[3]-4) y(x)+K[3] (27 K[3]-4)}+2}}{27 K[3]+27 y(x)-4}+\frac {9 i 2^{2/3} \sqrt {27 y(x)^2+(54 K[3]-4) y(x)+K[3] (27 K[3]-4)} \sqrt [3]{-27 K[3]-27 y(x)+3 \sqrt {3} \sqrt {27 y(x)^2+(54 K[3]-4) y(x)+K[3] (27 K[3]-4)}+2}}{27 K[3]+27 y(x)-4}+i 2^{2/3} \sqrt {3} \sqrt [3]{-27 K[3]-27 y(x)+3 \sqrt {3} \sqrt {27 y(x)^2+(54 K[3]-4) y(x)+K[3] (27 K[3]-4)}+2}-2^{2/3} \sqrt [3]{-27 K[3]-27 y(x)+3 \sqrt {3} \sqrt {27 y(x)^2+(54 K[3]-4) y(x)+K[3] (27 K[3]-4)}+2}+4\right )dK[3]+\int _1^{y(x)}\left (\frac {\sqrt [3]{-54 x-54 K[4]+6 \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)}+4} \left (-27 \sqrt {3} \sqrt [3]{-27 x-27 K[4]+3 \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)}+2} K[4]-27 i \sqrt [3]{-27 x-27 K[4]+3 \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)}+2} K[4]+54 i \sqrt [3]{2} K[4]+27 i x \left (i \sqrt {3} \sqrt [3]{-27 x-27 K[4]+3 \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)}+2}-\sqrt [3]{-27 x-27 K[4]+3 \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)}+2}+2 \sqrt [3]{2}\right )+6 i \sqrt [3]{2} \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)}-3 i \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)} \sqrt [3]{-27 x-27 K[4]+3 \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)}+2}-9 \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)} \sqrt [3]{-27 x-27 K[4]+3 \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)}+2}+4 \sqrt {3} \sqrt [3]{-27 x-27 K[4]+3 \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)}+2}+4 i \sqrt [3]{-27 x-27 K[4]+3 \sqrt {3} \sqrt {27 x^2+(54 K[4]-4) x+K[4] (27 K[4]-4)}+2}-8 i \sqrt [3]{2}\right )}{2 \left (-i+\sqrt {3}\right ) (27 x+27 K[4]-4)}-\int _1^x\frac {\sqrt [3]{2} \left (-27 i \sqrt {3} \sqrt [3]{-27 K[3]-27 K[4]+3 \sqrt {3} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)}+2} K[4]-81 \sqrt [3]{-27 K[3]-27 K[4]+3 \sqrt {3} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)}+2} K[4]+108 i \sqrt [3]{2} \sqrt {3} K[4]+27 i K[3] \left (-\sqrt {3} \sqrt [3]{-27 K[3]-27 K[4]+3 \sqrt {3} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)}+2}+3 i \sqrt [3]{-27 K[3]-27 K[4]+3 \sqrt {3} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)}+2}+4 \sqrt [3]{2} \sqrt {3}\right )-36 i \sqrt [3]{2} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)}+9 \sqrt {3} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)} \sqrt [3]{-27 K[3]-27 K[4]+3 \sqrt {3} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)}+2}+9 i \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)} \sqrt [3]{-27 K[3]-27 K[4]+3 \sqrt {3} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)}+2}-2 i \sqrt {3} \sqrt [3]{-27 K[3]-27 K[4]+3 \sqrt {3} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)}+2}-6 \sqrt [3]{-27 K[3]-27 K[4]+3 \sqrt {3} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)}+2}-4 i \sqrt [3]{2} \sqrt {3}\right )}{\left (-i+\sqrt {3}\right ) (27 K[3]+27 K[4]-4) \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)} \left (-27 K[3]-27 K[4]+3 \sqrt {3} \sqrt {27 K[3]^2+(54 K[4]-4) K[3]+K[4] (27 K[4]-4)}+2\right )^{2/3}}dK[3]\right )dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {1}{4} \left (\frac {2 \sqrt [3]{2} \left (-27 K[5]-27 y(x)+3 \sqrt {3} \sqrt {27 y(x)^2+(54 K[5]-4) y(x)+K[5] (27 K[5]-4)}+2\right )^{2/3} \left (-27 \sqrt {3} K[5]+27 i K[5]-27 \left (-i+\sqrt {3}\right ) y(x)+3 i \sqrt {3} \sqrt {27 y(x)^2+(54 K[5]-4) y(x)+K[5] (27 K[5]-4)}-9 \sqrt {27 y(x)^2+(54 K[5]-4) y(x)+K[5] (27 K[5]-4)}+4 \sqrt {3}-4 i\right )}{\left (i+\sqrt {3}\right ) (27 K[5]+27 y(x)-4)}-\frac {3\ 2^{2/3} \sqrt {3} \sqrt {27 y(x)^2+(54 K[5]-4) y(x)+K[5] (27 K[5]-4)} \sqrt [3]{-27 K[5]-27 y(x)+3 \sqrt {3} \sqrt {27 y(x)^2+(54 K[5]-4) y(x)+K[5] (27 K[5]-4)}+2}}{27 K[5]+27 y(x)-4}-\frac {9 i 2^{2/3} \sqrt {27 y(x)^2+(54 K[5]-4) y(x)+K[5] (27 K[5]-4)} \sqrt [3]{-27 K[5]-27 y(x)+3 \sqrt {3} \sqrt {27 y(x)^2+(54 K[5]-4) y(x)+K[5] (27 K[5]-4)}+2}}{27 K[5]+27 y(x)-4}-i 2^{2/3} \sqrt {3} \sqrt [3]{-27 K[5]-27 y(x)+3 \sqrt {3} \sqrt {27 y(x)^2+(54 K[5]-4) y(x)+K[5] (27 K[5]-4)}+2}-2^{2/3} \sqrt [3]{-27 K[5]-27 y(x)+3 \sqrt {3} \sqrt {27 y(x)^2+(54 K[5]-4) y(x)+K[5] (27 K[5]-4)}+2}+4\right )dK[5]+\int _1^{y(x)}\left (-\frac {\sqrt [3]{-54 x-54 K[6]+6 \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)}+4} \left (27 \sqrt {3} \sqrt [3]{-27 x-27 K[6]+3 \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)}+2} K[6]-27 i \sqrt [3]{-27 x-27 K[6]+3 \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)}+2} K[6]+54 i \sqrt [3]{2} K[6]+27 x \left (\sqrt {3} \sqrt [3]{-27 x-27 K[6]+3 \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)}+2}-i \sqrt [3]{-27 x-27 K[6]+3 \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)}+2}+2 i \sqrt [3]{2}\right )+6 i \sqrt [3]{2} \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)}-3 i \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)} \sqrt [3]{-27 x-27 K[6]+3 \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)}+2}+9 \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)} \sqrt [3]{-27 x-27 K[6]+3 \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)}+2}-4 \sqrt {3} \sqrt [3]{-27 x-27 K[6]+3 \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)}+2}+4 i \sqrt [3]{-27 x-27 K[6]+3 \sqrt {3} \sqrt {27 x^2+(54 K[6]-4) x+K[6] (27 K[6]-4)}+2}-8 i \sqrt [3]{2}\right )}{2 \left (i+\sqrt {3}\right ) (27 x+27 K[6]-4)}-\int _1^x\frac {\sqrt [3]{2} \left (27 i \sqrt {3} \sqrt [3]{-27 K[5]-27 K[6]+3 \sqrt {3} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)}+2} K[6]-81 \sqrt [3]{-27 K[5]-27 K[6]+3 \sqrt {3} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)}+2} K[6]-108 i \sqrt [3]{2} \sqrt {3} K[6]-27 i K[5] \left (-\sqrt {3} \sqrt [3]{-27 K[5]-27 K[6]+3 \sqrt {3} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)}+2}-3 i \sqrt [3]{-27 K[5]-27 K[6]+3 \sqrt {3} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)}+2}+4 \sqrt [3]{2} \sqrt {3}\right )+36 i \sqrt [3]{2} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)}+9 \sqrt {3} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)} \sqrt [3]{-27 K[5]-27 K[6]+3 \sqrt {3} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)}+2}-9 i \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)} \sqrt [3]{-27 K[5]-27 K[6]+3 \sqrt {3} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)}+2}+2 i \sqrt {3} \sqrt [3]{-27 K[5]-27 K[6]+3 \sqrt {3} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)}+2}-6 \sqrt [3]{-27 K[5]-27 K[6]+3 \sqrt {3} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)}+2}+4 i \sqrt [3]{2} \sqrt {3}\right )}{\left (i+\sqrt {3}\right ) (27 K[5]+27 K[6]-4) \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)} \left (-27 K[5]-27 K[6]+3 \sqrt {3} \sqrt {27 K[5]^2+(54 K[6]-4) K[5]+K[6] (27 K[6]-4)}+2\right )^{2/3}}dK[5]\right )dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.067 (sec), leaf count = 39
\[\left [y \left (x \right ) = -x +\frac {4}{27}, y \left (x \right ) = -\left (x -\textit {\_C1} \right )^{\frac {3}{2}}-\textit {\_C1}, y \left (x \right ) = \left (x -\textit {\_C1} \right )^{\frac {3}{2}}-\textit {\_C1}\right ]\] Mathematica raw input
DSolve[12*y'[x]^2 + 8*y'[x]^3 == 27*(x + y[x]),y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][(2 + 2^(2/3)*(2 - 27*K[1] - 27*y[x] + 3*Sqrt[3]*Sqrt[(K[1] + y[x])*(-4 + 27*(K[1] + y[x]))])^(1/3) + (3*2^(2/3)*Sqrt[3]*Sqrt[(K[1] + y[x])*(-4 + 27*(K[1] + y[x]))]*(2 - 27*K[1] - 27*y[x] + 3*Sqrt[3]*Sqrt[(K[1] + y[x])*(-4 + 27*(K[1] + y[x]))])^(1/3))/(-4 + 27*K[1] + 27*y[x]) + 2^(1/3)*(2 - 27*K[1] - 27*y[x] + 3*Sqrt[3]*Sqrt[(K[1] + y[x])*(-4 + 27*(K[1] + y[x]))])^(2/3) + (3*2^(1/3)*Sqrt[3]*Sqrt[(K[1] + y[x])*(-4 + 27*(K[1] + y[x]))]*(2 - 27*K[1] - 27*y[x] + 3*Sqrt[3]*Sqrt[(K[1] + y[x])*(-4 + 27*(K[1] + y[x]))])^(2/3))/(-4 + 27*K[1] + 27*y[x]))/2, {K[1], 1, x}] + Inactive[Integrate][((-4 + 27*x + 27*K[2] + 3*Sqrt[3]*Sqrt[27*x^2 + K[2]*(-4 + 27*K[2]) + x*(-4 + 54*K[2])])*(4 - 54*x - 54*K[2] + 6*Sqrt[3]*Sqrt[27*x^2 + K[2]*(-4 + 27*K[2]) + x*(-4 + 54*K[2])])^(1/3)*(2^(1/3) + (2 - 27*x - 27*K[2] + 3*Sqrt[3]*Sqrt[27*x^2 + K[2]*(-4 + 27*K[2]) + x*(-4 + 54*K[2])])^(1/3)))/(-8 + 54*x + 54*K[2]) - Inactive[Integrate][(2^(1/3)*(-2*2^(1/3)*Sqrt[3] + 54*2^(1/3)*Sqrt[3]*K[1] + 54*2^(1/3)*Sqrt[3]*K[2] - 18*2^(1/3)*Sqrt[(K[1] + K[2])*(-4 + 27*(K[1] + K[2]))] + 2*Sqrt[3]*(2 - 27*K[1] - 27*K[2] + 3*Sqrt[3]*Sqrt[(K[1] + K[2])*(-4 + 27*(K[1] + K[2]))])^(1/3) + 27*Sqrt[3]*K[1]*(2 - 27*K[1] - 27*K[2] + 3*Sqrt[3]*Sqrt[(K[1] + K[2])*(-4 + 27*(K[1] + K[2]))])^(1/3) + 27*Sqrt[3]*K[2]*(2 - 27*K[1] - 27*K[2] + 3*Sqrt[3]*Sqrt[(K[1] + K[2])*(-4 + 27*(K[1] + K[2]))])^(1/3) - 9*Sqrt[(K[1] + K[2])*(-4 + 27*(K[1] + K[2]))]*(2 - 27*K[1] - 27*K[2] + 3*Sqrt[3]*Sqrt[(K[1] + K[2])*(-4 + 27*(K[1] + K[2]))])^(1/3)))/((-4 + 27*K[1] + 27*K[2])*Sqrt[(K[1] + K[2])*(-4 + 27*(K[1] + K[2]))]*(2 - 27*K[1] - 27*K[2] + 3*Sqrt[3]*Sqrt[(K[1] + K[2])*(-4 + 27*(K[1] + K[2]))])^(2/3)), {K[1], 1, x}], {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(4 - 2^(2/3)*(2 - 27*K[3] - 27*y[x] + 3*Sqrt[3]*Sqrt[K[3]*(-4 + 27*K[3]) + (-4 + 54*K[3])*y[x] + 27*y[x]^2])^(1/3) + I*2^(2/3)*Sqrt[3]*(2 - 27*K[3] - 27*y[x] + 3*Sqrt[3]*Sqrt[K[3]*(-4 + 27*K[3]) + (-4 + 54*K[3])*y[x] + 27*y[x]^2])^(1/3) + ((9*I)*2^(2/3)*Sqrt[K[3]*(-4 + 27*K[3]) + (-4 + 54*K[3])*y[x] + 27*y[x]^2]*(2 - 27*K[3] - 27*y[x] + 3*Sqrt[3]*Sqrt[K[3]*(-4 + 27*K[3]) + (-4 + 54*K[3])*y[x] + 27*y[x]^2])^(1/3))/(-4 + 27*K[3] + 27*y[x]) - (3*2^(2/3)*Sqrt[3]*Sqrt[K[3]*(-4 + 27*K[3]) + (-4 + 54*K[3])*y[x] + 27*y[x]^2]*(2 - 27*K[3] - 27*y[x] + 3*Sqrt[3]*Sqrt[K[3]*(-4 + 27*K[3]) + (-4 + 54*K[3])*y[x] + 27*y[x]^2])^(1/3))/(-4 + 27*K[3] + 27*y[x]) + (2*2^(1/3)*(4*I + 4*Sqrt[3] - 27*(I + Sqrt[3])*K[3] - 27*(I + Sqrt[3])*y[x] - 9*Sqrt[K[3]*(-4 + 27*K[3]) + (-4 + 54*K[3])*y[x] + 27*y[x]^2] - (3*I)*Sqrt[3]*Sqrt[K[3]*(-4 + 27*K[3]) + (-4 + 54*K[3])*y[x] + 27*y[x]^2])*(2 - 27*K[3] - 27*y[x] + 3*Sqrt[3]*Sqrt[K[3]*(-4 + 27*K[3]) + (-4 + 54*K[3])*y[x] + 27*y[x]^2])^(2/3))/((-I + Sqrt[3])*(-4 + 27*K[3] + 27*y[x])))/4, {K[3], 1, x}] + Inactive[Integrate][((4 - 54*x - 54*K[4] + 6*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])])^(1/3)*((-8*I)*2^(1/3) + (54*I)*2^(1/3)*K[4] + (6*I)*2^(1/3)*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])] + (4*I)*(2 - 27*x - 27*K[4] + 3*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])])^(1/3) + 4*Sqrt[3]*(2 - 27*x - 27*K[4] + 3*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])])^(1/3) - (27*I)*K[4]*(2 - 27*x - 27*K[4] + 3*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])])^(1/3) - 27*Sqrt[3]*K[4]*(2 - 27*x - 27*K[4] + 3*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])])^(1/3) - 9*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])]*(2 - 27*x - 27*K[4] + 3*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])])^(1/3) - (3*I)*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])]*(2 - 27*x - 27*K[4] + 3*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])])^(1/3) + (27*I)*x*(2*2^(1/3) - (2 - 27*x - 27*K[4] + 3*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])])^(1/3) + I*Sqrt[3]*(2 - 27*x - 27*K[4] + 3*Sqrt[3]*Sqrt[27*x^2 + K[4]*(-4 + 27*K[4]) + x*(-4 + 54*K[4])])^(1/3))))/(2*(-I + Sqrt[3])*(-4 + 27*x + 27*K[4])) - Inactive[Integrate][(2^(1/3)*((-4*I)*2^(1/3)*Sqrt[3] + (108*I)*2^(1/3)*Sqrt[3]*K[4] - (36*I)*2^(1/3)*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])] - 6*(2 - 27*K[3] - 27*K[4] + 3*Sqrt[3]*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])])^(1/3) - (2*I)*Sqrt[3]*(2 - 27*K[3] - 27*K[4] + 3*Sqrt[3]*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])])^(1/3) - 81*K[4]*(2 - 27*K[3] - 27*K[4] + 3*Sqrt[3]*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])])^(1/3) - (27*I)*Sqrt[3]*K[4]*(2 - 27*K[3] - 27*K[4] + 3*Sqrt[3]*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])])^(1/3) + (9*I)*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])]*(2 - 27*K[3] - 27*K[4] + 3*Sqrt[3]*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])])^(1/3) + 9*Sqrt[3]*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])]*(2 - 27*K[3] - 27*K[4] + 3*Sqrt[3]*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])])^(1/3) + (27*I)*K[3]*(4*2^(1/3)*Sqrt[3] + (3*I)*(2 - 27*K[3] - 27*K[4] + 3*Sqrt[3]*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])])^(1/3) - Sqrt[3]*(2 - 27*K[3] - 27*K[4] + 3*Sqrt[3]*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])])^(1/3))))/((-I + Sqrt[3])*(-4 + 27*K[3] + 27*K[4])*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])]*(2 - 27*K[3] - 27*K[4] + 3*Sqrt[3]*Sqrt[27*K[3]^2 + K[4]*(-4 + 27*K[4]) + K[3]*(-4 + 54*K[4])])^(2/3)), {K[3], 1, x}], {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(4 - 2^(2/3)*(2 - 27*K[5] - 27*y[x] + 3*Sqrt[3]*Sqrt[K[5]*(-4 + 27*K[5]) + (-4 + 54*K[5])*y[x] + 27*y[x]^2])^(1/3) - I*2^(2/3)*Sqrt[3]*(2 - 27*K[5] - 27*y[x] + 3*Sqrt[3]*Sqrt[K[5]*(-4 + 27*K[5]) + (-4 + 54*K[5])*y[x] + 27*y[x]^2])^(1/3) - ((9*I)*2^(2/3)*Sqrt[K[5]*(-4 + 27*K[5]) + (-4 + 54*K[5])*y[x] + 27*y[x]^2]*(2 - 27*K[5] - 27*y[x] + 3*Sqrt[3]*Sqrt[K[5]*(-4 + 27*K[5]) + (-4 + 54*K[5])*y[x] + 27*y[x]^2])^(1/3))/(-4 + 27*K[5] + 27*y[x]) - (3*2^(2/3)*Sqrt[3]*Sqrt[K[5]*(-4 + 27*K[5]) + (-4 + 54*K[5])*y[x] + 27*y[x]^2]*(2 - 27*K[5] - 27*y[x] + 3*Sqrt[3]*Sqrt[K[5]*(-4 + 27*K[5]) + (-4 + 54*K[5])*y[x] + 27*y[x]^2])^(1/3))/(-4 + 27*K[5] + 27*y[x]) + (2*2^(1/3)*(-4*I + 4*Sqrt[3] + (27*I)*K[5] - 27*Sqrt[3]*K[5] - 27*(-I + Sqrt[3])*y[x] - 9*Sqrt[K[5]*(-4 + 27*K[5]) + (-4 + 54*K[5])*y[x] + 27*y[x]^2] + (3*I)*Sqrt[3]*Sqrt[K[5]*(-4 + 27*K[5]) + (-4 + 54*K[5])*y[x] + 27*y[x]^2])*(2 - 27*K[5] - 27*y[x] + 3*Sqrt[3]*Sqrt[K[5]*(-4 + 27*K[5]) + (-4 + 54*K[5])*y[x] + 27*y[x]^2])^(2/3))/((I + Sqrt[3])*(-4 + 27*K[5] + 27*y[x])))/4, {K[5], 1, x}] + Inactive[Integrate][-1/2*((4 - 54*x - 54*K[6] + 6*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])])^(1/3)*((-8*I)*2^(1/3) + (54*I)*2^(1/3)*K[6] + (6*I)*2^(1/3)*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])] + (4*I)*(2 - 27*x - 27*K[6] + 3*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])])^(1/3) - 4*Sqrt[3]*(2 - 27*x - 27*K[6] + 3*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])])^(1/3) - (27*I)*K[6]*(2 - 27*x - 27*K[6] + 3*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])])^(1/3) + 27*Sqrt[3]*K[6]*(2 - 27*x - 27*K[6] + 3*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])])^(1/3) + 9*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])]*(2 - 27*x - 27*K[6] + 3*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])])^(1/3) - (3*I)*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])]*(2 - 27*x - 27*K[6] + 3*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])])^(1/3) + 27*x*((2*I)*2^(1/3) - I*(2 - 27*x - 27*K[6] + 3*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])])^(1/3) + Sqrt[3]*(2 - 27*x - 27*K[6] + 3*Sqrt[3]*Sqrt[27*x^2 + K[6]*(-4 + 27*K[6]) + x*(-4 + 54*K[6])])^(1/3))))/((I + Sqrt[3])*(-4 + 27*x + 27*K[6])) - Inactive[Integrate][(2^(1/3)*((4*I)*2^(1/3)*Sqrt[3] - (108*I)*2^(1/3)*Sqrt[3]*K[6] + (36*I)*2^(1/3)*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])] - 6*(2 - 27*K[5] - 27*K[6] + 3*Sqrt[3]*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])])^(1/3) + (2*I)*Sqrt[3]*(2 - 27*K[5] - 27*K[6] + 3*Sqrt[3]*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])])^(1/3) - 81*K[6]*(2 - 27*K[5] - 27*K[6] + 3*Sqrt[3]*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])])^(1/3) + (27*I)*Sqrt[3]*K[6]*(2 - 27*K[5] - 27*K[6] + 3*Sqrt[3]*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])])^(1/3) - (9*I)*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])]*(2 - 27*K[5] - 27*K[6] + 3*Sqrt[3]*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])])^(1/3) + 9*Sqrt[3]*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])]*(2 - 27*K[5] - 27*K[6] + 3*Sqrt[3]*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])])^(1/3) - (27*I)*K[5]*(4*2^(1/3)*Sqrt[3] - (3*I)*(2 - 27*K[5] - 27*K[6] + 3*Sqrt[3]*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])])^(1/3) - Sqrt[3]*(2 - 27*K[5] - 27*K[6] + 3*Sqrt[3]*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])])^(1/3))))/((I + Sqrt[3])*(-4 + 27*K[5] + 27*K[6])*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])]*(2 - 27*K[5] - 27*K[6] + 3*Sqrt[3]*Sqrt[27*K[5]^2 + K[6]*(-4 + 27*K[6]) + K[5]*(-4 + 54*K[6])])^(2/3)), {K[5], 1, x}], {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(8*diff(y(x),x)^3+12*diff(y(x),x)^2 = 27*x+27*y(x), y(x))
Maple raw output
[y(x) = -x+4/27, y(x) = -(x-_C1)^(3/2)-_C1, y(x) = (x-_C1)^(3/2)-_C1]