ODE
\[ x^4 y'(x)^3-x^3 y(x) y'(x)^2-x^2 y(x)^2 y'(x)+x y(x)^3=1 \] ODE Classification
[[_1st_order, _with_linear_symmetries]]
Book solution method
Change of variable
Mathematica ✓
cpu = 162.666 (sec), leaf count = 3049
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {1024 y(x)^5 K[1]^{15}-864 y(x)^2 K[1]^{14}-64\ 2^{2/3} y(x)^4 \sqrt [3]{-16 y(x)^3 K[1]^9+27 K[1]^8+3 \sqrt {3} \sqrt {-K[1]^{16} \left (32 K[1] y(x)^3-27\right )}} K[1]^{12}+32 \sqrt [3]{2} y(x)^3 \left (-16 y(x)^3 K[1]^9+27 K[1]^8+3 \sqrt {3} \sqrt {-K[1]^{16} \left (32 K[1] y(x)^3-27\right )}\right )^{2/3} K[1]^9-6\ 2^{2/3} y(x) \left (\sqrt {3} \sqrt {-K[1]^{16} \left (32 K[1] y(x)^3-27\right )}-9 K[1]^8\right ) \sqrt [3]{-16 y(x)^3 K[1]^9+27 K[1]^8+3 \sqrt {3} \sqrt {-K[1]^{16} \left (32 K[1] y(x)^3-27\right )}} K[1]^3+3 \sqrt [3]{2} \left (\sqrt {3} \sqrt {-K[1]^{16} \left (32 K[1] y(x)^3-27\right )}-9 K[1]^8\right ) \left (-16 y(x)^3 K[1]^9+27 K[1]^8+3 \sqrt {3} \sqrt {-K[1]^{16} \left (32 K[1] y(x)^3-27\right )}\right )^{2/3}}{32 K[1]^{15} y(x)^2 \left (32 K[1] y(x)^3-27\right )}dK[1]+\int _1^{y(x)}\left (\frac {3 \left (32 K[2]^3 x^9-27 x^8+3 \sqrt {3} \sqrt {-x^{16} \left (32 x K[2]^3-27\right )}\right ) \sqrt [3]{-32 K[2]^3 x^9+54 x^8+6 \sqrt {3} \sqrt {-x^{16} \left (32 x K[2]^3-27\right )}} \left (\sqrt [3]{-16 K[2]^3 x^9+27 x^8+3 \sqrt {3} \sqrt {-x^{16} \left (32 x K[2]^3-27\right )}}-2 \sqrt [3]{2} x^3 K[2]\right )}{32 x^{14} K[2]^3 \left (32 x K[2]^3-27\right )}-\int _1^x-\frac {3 K[1]^{18} \left (32 \sqrt [3]{2} \sqrt {3} K[2]^4 K[1]^{12}-108 \sqrt [3]{2} \sqrt {3} K[2] K[1]^{11}+16 \sqrt {3} K[2]^3 \sqrt [3]{-16 K[2]^3 K[1]^9+27 K[1]^8+3 \sqrt {3} \sqrt {-K[1]^{16} \left (32 K[1] K[2]^3-27\right )}} K[1]^9+27 \sqrt {3} \sqrt [3]{-16 K[2]^3 K[1]^9+27 K[1]^8+3 \sqrt {3} \sqrt {-K[1]^{16} \left (32 K[1] K[2]^3-27\right )}} K[1]^8-36 \sqrt [3]{2} K[2] \sqrt {-K[1]^{16} \left (32 K[1] K[2]^3-27\right )} K[1]^3+9 \sqrt {-K[1]^{16} \left (32 K[1] K[2]^3-27\right )} \sqrt [3]{-16 K[2]^3 K[1]^9+27 K[1]^8+3 \sqrt {3} \sqrt {-K[1]^{16} \left (32 K[1] K[2]^3-27\right )}}\right )}{\left (-K[1]^{16} \left (32 K[1] K[2]^3-27\right )\right )^{3/2} \left (-32 K[2]^3 K[1]^9+54 K[1]^8+6 \sqrt {3} \sqrt {-K[1]^{16} \left (32 K[1] K[2]^3-27\right )}\right )^{2/3}}dK[1]\right )dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {1024 \left (i+\sqrt {3}\right ) y(x)^5 K[3]^{15}-864 \left (i+\sqrt {3}\right ) y(x)^2 K[3]^{14}+128 i 2^{2/3} y(x)^4 \sqrt [3]{-16 y(x)^3 K[3]^9+27 K[3]^8+3 \sqrt {3} \sqrt {-K[3]^{16} \left (32 K[3] y(x)^3-27\right )}} K[3]^{12}-32 \sqrt [3]{2} \left (-i+\sqrt {3}\right ) y(x)^3 \left (-16 y(x)^3 K[3]^9+27 K[3]^8+3 \sqrt {3} \sqrt {-K[3]^{16} \left (32 K[3] y(x)^3-27\right )}\right )^{2/3} K[3]^9+12 i 2^{2/3} y(x) \left (\sqrt {3} \sqrt {-K[3]^{16} \left (32 K[3] y(x)^3-27\right )}-9 K[3]^8\right ) \sqrt [3]{-16 y(x)^3 K[3]^9+27 K[3]^8+3 \sqrt {3} \sqrt {-K[3]^{16} \left (32 K[3] y(x)^3-27\right )}} K[3]^3+3 \sqrt [3]{2} \left (-16 y(x)^3 K[3]^9+27 K[3]^8+3 \sqrt {3} \sqrt {-K[3]^{16} \left (32 K[3] y(x)^3-27\right )}\right )^{2/3} \left (9 \left (-i+\sqrt {3}\right ) K[3]^8+i \left (3 i+\sqrt {3}\right ) \sqrt {-K[3]^{16} \left (32 K[3] y(x)^3-27\right )}\right )}{32 \left (i+\sqrt {3}\right ) K[3]^{15} y(x)^2 \left (32 K[3] y(x)^3-27\right )}dK[3]+\int _1^{y(x)}\left (-\frac {3 \sqrt [3]{-32 K[4]^3 x^9+54 x^8+6 \sqrt {3} \sqrt {-x^{16} \left (32 x K[4]^3-27\right )}} \left (12 i \sqrt [3]{2} \sqrt {3} K[4] x^{11}+3 i \left (3 i+\sqrt {3}\right ) \sqrt [3]{-16 K[4]^3 x^9+27 x^8+3 \sqrt {3} \sqrt {-x^{16} \left (32 x K[4]^3-27\right )}} x^8-4 i \sqrt [3]{2} K[4] \sqrt {-x^{16} \left (32 x K[4]^3-27\right )} x^3+\left (-i+\sqrt {3}\right ) \sqrt {-x^{16} \left (32 x K[4]^3-27\right )} \sqrt [3]{-16 K[4]^3 x^9+27 x^8+3 \sqrt {3} \sqrt {-x^{16} \left (32 x K[4]^3-27\right )}}\right )}{32 \left (i+\sqrt {3}\right ) x^6 K[4]^3 \sqrt {-x^{16} \left (32 x K[4]^3-27\right )}}-\int _1^x\frac {3 i K[3]^{18} \left (64 \sqrt [3]{2} \sqrt {3} K[4]^4 K[3]^{12}-216 \sqrt [3]{2} \sqrt {3} K[4] K[3]^{11}-16 \left (3 i+\sqrt {3}\right ) K[4]^3 \sqrt [3]{-16 K[4]^3 K[3]^9+27 K[3]^8+3 \sqrt {3} \sqrt {-K[3]^{16} \left (32 K[3] K[4]^3-27\right )}} K[3]^9-27 \left (3 i+\sqrt {3}\right ) \sqrt [3]{-16 K[4]^3 K[3]^9+27 K[3]^8+3 \sqrt {3} \sqrt {-K[3]^{16} \left (32 K[3] K[4]^3-27\right )}} K[3]^8-72 \sqrt [3]{2} K[4] \sqrt {-K[3]^{16} \left (32 K[3] K[4]^3-27\right )} K[3]^3-9 i \left (-i+\sqrt {3}\right ) \sqrt {-K[3]^{16} \left (32 K[3] K[4]^3-27\right )} \sqrt [3]{-16 K[4]^3 K[3]^9+27 K[3]^8+3 \sqrt {3} \sqrt {-K[3]^{16} \left (32 K[3] K[4]^3-27\right )}}\right )}{\left (i+\sqrt {3}\right ) \left (-K[3]^{16} \left (32 K[3] K[4]^3-27\right )\right )^{3/2} \left (-32 K[4]^3 K[3]^9+54 K[3]^8+6 \sqrt {3} \sqrt {-K[3]^{16} \left (32 K[3] K[4]^3-27\right )}\right )^{2/3}}dK[3]\right )dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x-\frac {-1024 \left (-i+\sqrt {3}\right ) y(x)^5 K[5]^{15}+864 \left (-i+\sqrt {3}\right ) y(x)^2 K[5]^{14}+128 i 2^{2/3} y(x)^4 \sqrt [3]{-16 y(x)^3 K[5]^9+27 K[5]^8+3 \sqrt {3} \sqrt {-K[5]^{16} \left (32 K[5] y(x)^3-27\right )}} K[5]^{12}+32 \sqrt [3]{2} \left (i+\sqrt {3}\right ) y(x)^3 \left (-16 y(x)^3 K[5]^9+27 K[5]^8+3 \sqrt {3} \sqrt {-K[5]^{16} \left (32 K[5] y(x)^3-27\right )}\right )^{2/3} K[5]^9+12 i 2^{2/3} y(x) \left (\sqrt {3} \sqrt {-K[5]^{16} \left (32 K[5] y(x)^3-27\right )}-9 K[5]^8\right ) \sqrt [3]{-16 y(x)^3 K[5]^9+27 K[5]^8+3 \sqrt {3} \sqrt {-K[5]^{16} \left (32 K[5] y(x)^3-27\right )}} K[5]^3+3 \sqrt [3]{2} \left (-16 y(x)^3 K[5]^9+27 K[5]^8+3 \sqrt {3} \sqrt {-K[5]^{16} \left (32 K[5] y(x)^3-27\right )}\right )^{2/3} \left (\left (3+i \sqrt {3}\right ) \sqrt {-K[5]^{16} \left (32 K[5] y(x)^3-27\right )}-9 \left (i+\sqrt {3}\right ) K[5]^8\right )}{32 \left (-i+\sqrt {3}\right ) K[5]^{15} y(x)^2 \left (32 K[5] y(x)^3-27\right )}dK[5]+\int _1^{y(x)}\left (\frac {3 \sqrt [3]{-32 K[6]^3 x^9+54 x^8+6 \sqrt {3} \sqrt {-x^{16} \left (32 x K[6]^3-27\right )}} \left (12 i \sqrt [3]{2} \sqrt {3} K[6] x^{11}+3 \left (3+i \sqrt {3}\right ) \sqrt [3]{-16 K[6]^3 x^9+27 x^8+3 \sqrt {3} \sqrt {-x^{16} \left (32 x K[6]^3-27\right )}} x^8-4 i \sqrt [3]{2} K[6] \sqrt {-x^{16} \left (32 x K[6]^3-27\right )} x^3-\left (i+\sqrt {3}\right ) \sqrt {-x^{16} \left (32 x K[6]^3-27\right )} \sqrt [3]{-16 K[6]^3 x^9+27 x^8+3 \sqrt {3} \sqrt {-x^{16} \left (32 x K[6]^3-27\right )}}\right )}{32 \left (-i+\sqrt {3}\right ) x^6 K[6]^3 \sqrt {-x^{16} \left (32 x K[6]^3-27\right )}}-\int _1^x-\frac {3 i K[5]^{18} \left (64 \sqrt [3]{2} \sqrt {3} K[6]^4 K[5]^{12}-216 \sqrt [3]{2} \sqrt {3} K[6] K[5]^{11}-16 \left (-3 i+\sqrt {3}\right ) K[6]^3 \sqrt [3]{-16 K[6]^3 K[5]^9+27 K[5]^8+3 \sqrt {3} \sqrt {-K[5]^{16} \left (32 K[5] K[6]^3-27\right )}} K[5]^9-27 \left (-3 i+\sqrt {3}\right ) \sqrt [3]{-16 K[6]^3 K[5]^9+27 K[5]^8+3 \sqrt {3} \sqrt {-K[5]^{16} \left (32 K[5] K[6]^3-27\right )}} K[5]^8-72 \sqrt [3]{2} K[6] \sqrt {-K[5]^{16} \left (32 K[5] K[6]^3-27\right )} K[5]^3+9 i \left (i+\sqrt {3}\right ) \sqrt {-K[5]^{16} \left (32 K[5] K[6]^3-27\right )} \sqrt [3]{-16 K[6]^3 K[5]^9+27 K[5]^8+3 \sqrt {3} \sqrt {-K[5]^{16} \left (32 K[5] K[6]^3-27\right )}}\right )}{\left (-i+\sqrt {3}\right ) \left (-K[5]^{16} \left (32 K[5] K[6]^3-27\right )\right )^{3/2} \left (-32 K[6]^3 K[5]^9+54 K[5]^8+6 \sqrt {3} \sqrt {-K[5]^{16} \left (32 K[5] K[6]^3-27\right )}\right )^{2/3}}dK[5]\right )dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 23.634 (sec), leaf count = 544
\[\left [y \left (x \right ) = \frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4 x}, y \left (x \right ) = -\frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{8 x}-\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{8 x}, y \left (x \right ) = -\frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{8 x}+\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{8 x}, y \left (x \right ) = \frac {\RootOf \left (-\ln \left (x \right )+3 \left (\int _{}^{\textit {\_Z}}\frac {\left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {1}{3}}}{4 \,2^{\frac {1}{3}} \textit {\_a}^{2}+2 \left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {1}{3}} \textit {\_a} +\left (128 \textit {\_a}^{6}-48 \textit {\_a}^{3} \sqrt {-96 \textit {\_a}^{3}+81}-864 \textit {\_a}^{3}+81 \sqrt {-96 \textit {\_a}^{3}+81}+729\right )^{\frac {1}{3}}}d \textit {\_a} \right )+\textit {\_C1} \right )}{x^{\frac {1}{3}}}, y \left (x \right ) = \frac {\RootOf \left (-\ln \left (x \right )+6 \left (\int _{}^{\textit {\_Z}}\frac {\left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {1}{3}}}{4 i 2^{\frac {1}{3}} \sqrt {3}\, \textit {\_a}^{2}-4 \,2^{\frac {1}{3}} \textit {\_a}^{2}-i \left (128 \textit {\_a}^{6}-48 \textit {\_a}^{3} \sqrt {-96 \textit {\_a}^{3}+81}-864 \textit {\_a}^{3}+81 \sqrt {-96 \textit {\_a}^{3}+81}+729\right )^{\frac {1}{3}} \sqrt {3}+4 \left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {1}{3}} \textit {\_a} -\left (128 \textit {\_a}^{6}-48 \textit {\_a}^{3} \sqrt {-96 \textit {\_a}^{3}+81}-864 \textit {\_a}^{3}+81 \sqrt {-96 \textit {\_a}^{3}+81}+729\right )^{\frac {1}{3}}}d \textit {\_a} \right )+\textit {\_C1} \right )}{x^{\frac {1}{3}}}, y \left (x \right ) = \frac {\RootOf \left (-\ln \left (x \right )+6 \left (\int _{}^{\textit {\_Z}}-\frac {\left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {1}{3}}}{4 i 2^{\frac {1}{3}} \sqrt {3}\, \textit {\_a}^{2}+4 \,2^{\frac {1}{3}} \textit {\_a}^{2}-i \left (128 \textit {\_a}^{6}-48 \textit {\_a}^{3} \sqrt {-96 \textit {\_a}^{3}+81}-864 \textit {\_a}^{3}+81 \sqrt {-96 \textit {\_a}^{3}+81}+729\right )^{\frac {1}{3}} \sqrt {3}-4 \left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {1}{3}} \textit {\_a} +\left (128 \textit {\_a}^{6}-48 \textit {\_a}^{3} \sqrt {-96 \textit {\_a}^{3}+81}-864 \textit {\_a}^{3}+81 \sqrt {-96 \textit {\_a}^{3}+81}+729\right )^{\frac {1}{3}}}d \textit {\_a} \right )+\textit {\_C1} \right )}{x^{\frac {1}{3}}}\right ]\] Mathematica raw input
DSolve[x*y[x]^3 - x^2*y[x]^2*y'[x] - x^3*y[x]*y'[x]^2 + x^4*y'[x]^3 == 1,y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][(-864*K[1]^14*y[x]^2 + 1024*K[1]^15*y[x]^5 - 64*2^(2/3)*K[1]^12*y[x]^4*(27*K[1]^8 - 16*K[1]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*y[x]^3))])^(1/3) - 6*2^(2/3)*K[1]^3*y[x]*(-9*K[1]^8 + Sqrt[3]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*y[x]^3))])*(27*K[1]^8 - 16*K[1]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*y[x]^3))])^(1/3) + 32*2^(1/3)*K[1]^9*y[x]^3*(27*K[1]^8 - 16*K[1]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*y[x]^3))])^(2/3) + 3*2^(1/3)*(-9*K[1]^8 + Sqrt[3]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*y[x]^3))])*(27*K[1]^8 - 16*K[1]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*y[x]^3))])^(2/3))/(32*K[1]^15*y[x]^2*(-27 + 32*K[1]*y[x]^3)), {K[1], 1, x}] + Inactive[Integrate][(3*(-27*x^8 + 32*x^9*K[2]^3 + 3*Sqrt[3]*Sqrt[-(x^16*(-27 + 32*x*K[2]^3))])*(54*x^8 - 32*x^9*K[2]^3 + 6*Sqrt[3]*Sqrt[-(x^16*(-27 + 32*x*K[2]^3))])^(1/3)*(-2*2^(1/3)*x^3*K[2] + (27*x^8 - 16*x^9*K[2]^3 + 3*Sqrt[3]*Sqrt[-(x^16*(-27 + 32*x*K[2]^3))])^(1/3)))/(32*x^14*K[2]^3*(-27 + 32*x*K[2]^3)) - Inactive[Integrate][(-3*K[1]^18*(-108*2^(1/3)*Sqrt[3]*K[1]^11*K[2] + 32*2^(1/3)*Sqrt[3]*K[1]^12*K[2]^4 - 36*2^(1/3)*K[1]^3*K[2]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*K[2]^3))] + 27*Sqrt[3]*K[1]^8*(27*K[1]^8 - 16*K[1]^9*K[2]^3 + 3*Sqrt[3]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*K[2]^3))])^(1/3) + 16*Sqrt[3]*K[1]^9*K[2]^3*(27*K[1]^8 - 16*K[1]^9*K[2]^3 + 3*Sqrt[3]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*K[2]^3))])^(1/3) + 9*Sqrt[-(K[1]^16*(-27 + 32*K[1]*K[2]^3))]*(27*K[1]^8 - 16*K[1]^9*K[2]^3 + 3*Sqrt[3]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*K[2]^3))])^(1/3)))/((-(K[1]^16*(-27 + 32*K[1]*K[2]^3)))^(3/2)*(54*K[1]^8 - 32*K[1]^9*K[2]^3 + 6*Sqrt[3]*Sqrt[-(K[1]^16*(-27 + 32*K[1]*K[2]^3))])^(2/3)), {K[1], 1, x}], {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(-864*(I + Sqrt[3])*K[3]^14*y[x]^2 + 1024*(I + Sqrt[3])*K[3]^15*y[x]^5 + (128*I)*2^(2/3)*K[3]^12*y[x]^4*(27*K[3]^8 - 16*K[3]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^16*(-27 + 32*K[3]*y[x]^3))])^(1/3) + (12*I)*2^(2/3)*K[3]^3*y[x]*(-9*K[3]^8 + Sqrt[3]*Sqrt[-(K[3]^16*(-27 + 32*K[3]*y[x]^3))])*(27*K[3]^8 - 16*K[3]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^16*(-27 + 32*K[3]*y[x]^3))])^(1/3) - 32*2^(1/3)*(-I + Sqrt[3])*K[3]^9*y[x]^3*(27*K[3]^8 - 16*K[3]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^16*(-27 + 32*K[3]*y[x]^3))])^(2/3) + 3*2^(1/3)*(27*K[3]^8 - 16*K[3]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^16*(-27 + 32*K[3]*y[x]^3))])^(2/3)*(9*(-I + Sqrt[3])*K[3]^8 + I*(3*I + Sqrt[3])*Sqrt[-(K[3]^16*(-27 + 32*K[3]*y[x]^3))]))/(32*(I + Sqrt[3])*K[3]^15*y[x]^2*(-27 + 32*K[3]*y[x]^3)), {K[3], 1, x}] + Inactive[Integrate][(-3*(54*x^8 - 32*x^9*K[4]^3 + 6*Sqrt[3]*Sqrt[-(x^16*(-27 + 32*x*K[4]^3))])^(1/3)*((12*I)*2^(1/3)*Sqrt[3]*x^11*K[4] - (4*I)*2^(1/3)*x^3*K[4]*Sqrt[-(x^16*(-27 + 32*x*K[4]^3))] + (3*I)*(3*I + Sqrt[3])*x^8*(27*x^8 - 16*x^9*K[4]^3 + 3*Sqrt[3]*Sqrt[-(x^16*(-27 + 32*x*K[4]^3))])^(1/3) + (-I + Sqrt[3])*Sqrt[-(x^16*(-27 + 32*x*K[4]^3))]*(27*x^8 - 16*x^9*K[4]^3 + 3*Sqrt[3]*Sqrt[-(x^16*(-27 + 32*x*K[4]^3))])^(1/3)))/(32*(I + Sqrt[3])*x^6*K[4]^3*Sqrt[-(x^16*(-27 + 32*x*K[4]^3))]) - Inactive[Integrate][((3*I)*K[3]^18*(-216*2^(1/3)*Sqrt[3]*K[3]^11*K[4] + 64*2^(1/3)*Sqrt[3]*K[3]^12*K[4]^4 - 72*2^(1/3)*K[3]^3*K[4]*Sqrt[-(K[3]^16*(-27 + 32*K[3]*K[4]^3))] - 27*(3*I + Sqrt[3])*K[3]^8*(27*K[3]^8 - 16*K[3]^9*K[4]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^16*(-27 + 32*K[3]*K[4]^3))])^(1/3) - 16*(3*I + Sqrt[3])*K[3]^9*K[4]^3*(27*K[3]^8 - 16*K[3]^9*K[4]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^16*(-27 + 32*K[3]*K[4]^3))])^(1/3) - (9*I)*(-I + Sqrt[3])*Sqrt[-(K[3]^16*(-27 + 32*K[3]*K[4]^3))]*(27*K[3]^8 - 16*K[3]^9*K[4]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^16*(-27 + 32*K[3]*K[4]^3))])^(1/3)))/((I + Sqrt[3])*(-(K[3]^16*(-27 + 32*K[3]*K[4]^3)))^(3/2)*(54*K[3]^8 - 32*K[3]^9*K[4]^3 + 6*Sqrt[3]*Sqrt[-(K[3]^16*(-27 + 32*K[3]*K[4]^3))])^(2/3)), {K[3], 1, x}], {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][-1/32*(864*(-I + Sqrt[3])*K[5]^14*y[x]^2 - 1024*(-I + Sqrt[3])*K[5]^15*y[x]^5 + (128*I)*2^(2/3)*K[5]^12*y[x]^4*(27*K[5]^8 - 16*K[5]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[5]^16*(-27 + 32*K[5]*y[x]^3))])^(1/3) + (12*I)*2^(2/3)*K[5]^3*y[x]*(-9*K[5]^8 + Sqrt[3]*Sqrt[-(K[5]^16*(-27 + 32*K[5]*y[x]^3))])*(27*K[5]^8 - 16*K[5]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[5]^16*(-27 + 32*K[5]*y[x]^3))])^(1/3) + 32*2^(1/3)*(I + Sqrt[3])*K[5]^9*y[x]^3*(27*K[5]^8 - 16*K[5]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[5]^16*(-27 + 32*K[5]*y[x]^3))])^(2/3) + 3*2^(1/3)*(27*K[5]^8 - 16*K[5]^9*y[x]^3 + 3*Sqrt[3]*Sqrt[-(K[5]^16*(-27 + 32*K[5]*y[x]^3))])^(2/3)*(-9*(I + Sqrt[3])*K[5]^8 + (3 + I*Sqrt[3])*Sqrt[-(K[5]^16*(-27 + 32*K[5]*y[x]^3))]))/((-I + Sqrt[3])*K[5]^15*y[x]^2*(-27 + 32*K[5]*y[x]^3)), {K[5], 1, x}] + Inactive[Integrate][(3*(54*x^8 - 32*x^9*K[6]^3 + 6*Sqrt[3]*Sqrt[-(x^16*(-27 + 32*x*K[6]^3))])^(1/3)*((12*I)*2^(1/3)*Sqrt[3]*x^11*K[6] - (4*I)*2^(1/3)*x^3*K[6]*Sqrt[-(x^16*(-27 + 32*x*K[6]^3))] + 3*(3 + I*Sqrt[3])*x^8*(27*x^8 - 16*x^9*K[6]^3 + 3*Sqrt[3]*Sqrt[-(x^16*(-27 + 32*x*K[6]^3))])^(1/3) - (I + Sqrt[3])*Sqrt[-(x^16*(-27 + 32*x*K[6]^3))]*(27*x^8 - 16*x^9*K[6]^3 + 3*Sqrt[3]*Sqrt[-(x^16*(-27 + 32*x*K[6]^3))])^(1/3)))/(32*(-I + Sqrt[3])*x^6*K[6]^3*Sqrt[-(x^16*(-27 + 32*x*K[6]^3))]) - Inactive[Integrate][((-3*I)*K[5]^18*(-216*2^(1/3)*Sqrt[3]*K[5]^11*K[6] + 64*2^(1/3)*Sqrt[3]*K[5]^12*K[6]^4 - 72*2^(1/3)*K[5]^3*K[6]*Sqrt[-(K[5]^16*(-27 + 32*K[5]*K[6]^3))] - 27*(-3*I + Sqrt[3])*K[5]^8*(27*K[5]^8 - 16*K[5]^9*K[6]^3 + 3*Sqrt[3]*Sqrt[-(K[5]^16*(-27 + 32*K[5]*K[6]^3))])^(1/3) - 16*(-3*I + Sqrt[3])*K[5]^9*K[6]^3*(27*K[5]^8 - 16*K[5]^9*K[6]^3 + 3*Sqrt[3]*Sqrt[-(K[5]^16*(-27 + 32*K[5]*K[6]^3))])^(1/3) + (9*I)*(I + Sqrt[3])*Sqrt[-(K[5]^16*(-27 + 32*K[5]*K[6]^3))]*(27*K[5]^8 - 16*K[5]^9*K[6]^3 + 3*Sqrt[3]*Sqrt[-(K[5]^16*(-27 + 32*K[5]*K[6]^3))])^(1/3)))/((-I + Sqrt[3])*(-(K[5]^16*(-27 + 32*K[5]*K[6]^3)))^(3/2)*(54*K[5]^8 - 32*K[5]^9*K[6]^3 + 6*Sqrt[3]*Sqrt[-(K[5]^16*(-27 + 32*K[5]*K[6]^3))])^(2/3)), {K[5], 1, x}], {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(x^4*diff(y(x),x)^3-x^3*y(x)*diff(y(x),x)^2-x^2*y(x)^2*diff(y(x),x)+x*y(x)^3 = 1, y(x))
Maple raw output
[y(x) = 3/4/x*2^(1/3)*(x^2)^(1/3), y(x) = -3/8/x*2^(1/3)*(x^2)^(1/3)-3/8*I*3^(1/2)/x*2^(1/3)*(x^2)^(1/3), y(x) = -3/8/x*2^(1/3)*(x^2)^(1/3)+3/8*I*3^(1/2)/x*2^(1/3)*(x^2)^(1/3), y(x) = RootOf(-ln(x)+3*Intat((-16*_a^3+3*(-96*_a^3+81)^(1/2)+27)^(1/3)/(4*2^(1/3)*_a^2+2*(-16*_a^3+3*(-96*_a^3+81)^(1/2)+27)^(1/3)*_a+(128*_a^6-48*_a^3*(-96*_a^3+81)^(1/2)-864*_a^3+81*(-96*_a^3+81)^(1/2)+729)^(1/3)),_a = _Z)+_C1)/x^(1/3), y(x) = RootOf(-ln(x)+6*Intat((-16*_a^3+3*(-96*_a^3+81)^(1/2)+27)^(1/3)/(4*I*2^(1/3)*3^(1/2)*_a^2-4*2^(1/3)*_a^2-I*(128*_a^6-48*_a^3*(-96*_a^3+81)^(1/2)-864*_a^3+81*(-96*_a^3+81)^(1/2)+729)^(1/3)*3^(1/2)+4*(-16*_a^3+3*(-96*_a^3+81)^(1/2)+27)^(1/3)*_a-(128*_a^6-48*_a^3*(-96*_a^3+81)^(1/2)-864*_a^3+81*(-96*_a^3+81)^(1/2)+729)^(1/3)),_a = _Z)+_C1)/x^(1/3), y(x) = RootOf(-ln(x)+6*Intat(-(-16*_a^3+3*(-96*_a^3+81)^(1/2)+27)^(1/3)/(4*I*2^(1/3)*3^(1/2)*_a^2+4*2^(1/3)*_a^2-I*(128*_a^6-48*_a^3*(-96*_a^3+81)^(1/2)-864*_a^3+81*(-96*_a^3+81)^(1/2)+729)^(1/3)*3^(1/2)-4*(-16*_a^3+3*(-96*_a^3+81)^(1/2)+27)^(1/3)*_a+(128*_a^6-48*_a^3*(-96*_a^3+81)^(1/2)-864*_a^3+81*(-96*_a^3+81)^(1/2)+729)^(1/3)),_a = _Z)+_C1)/x^(1/3)]