ODE
\[ 2 y(x) y'(x)^3+3 y(x) y'(x)+x=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 58.7036 (sec), leaf count = 3587
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {\left (4 K[1]-2 \sqrt [3]{2 \sqrt {y(x)^4 \left (K[1]^2+2 y(x)^2\right )}-2 K[1] y(x)^2}\right ) y(x)^4+K[1]^2 \left (2 K[1]-\sqrt [3]{2 \sqrt {y(x)^4 \left (K[1]^2+2 y(x)^2\right )}-2 K[1] y(x)^2}\right ) y(x)^2+\sqrt {y(x)^4 \left (K[1]^2+2 y(x)^2\right )} \sqrt [3]{2 \sqrt {y(x)^4 \left (K[1]^2+2 y(x)^2\right )}-2 K[1] y(x)^2} \left (\sqrt [3]{2 \sqrt {y(x)^4 \left (K[1]^2+2 y(x)^2\right )}-2 K[1] y(x)^2}-K[1]\right )}{2 y(x)^2 \left (K[1]^2+y(x)^2\right ) \left (K[1]^2+2 y(x)^2\right )}dK[1]+\int _1^{y(x)}\frac {4 K[2]^6-2 \left (x^4+3 K[2]^2 x^2+2 K[2]^4\right ) \int _1^x\frac {K[2]^5 \left (6 \sqrt [3]{2} K[2]^8-3\ 2^{2/3} \sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )} \sqrt [3]{\sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )}-K[1] K[2]^2} K[2]^4-2\ 2^{2/3} K[1]^5 \sqrt [3]{\sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )}-K[1] K[2]^2} K[2]^2+2 K[1]^4 \left (2^{2/3} \sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )} \sqrt [3]{\sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )}-K[1] K[2]^2}-3 \sqrt [3]{2} K[2]^4\right )+2 K[1]^2 \left (2^{2/3} K[2]^2 \sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )} \sqrt [3]{\sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )}-K[1] K[2]^2}-3 \sqrt [3]{2} K[2]^6\right )+2 K[1]^3 \sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )} \left (\sqrt [3]{2} K[2]^2-3 \left (\sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )}-K[1] K[2]^2\right )^{2/3}\right )+K[1] \left (5\ 2^{2/3} \sqrt [3]{\sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )}-K[1] K[2]^2} K[2]^6+2 \sqrt [3]{2} \sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )} K[2]^4-12 \sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )} \left (\sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )}-K[1] K[2]^2\right )^{2/3} K[2]^2\right )\right )}{3 \left (K[1]^2+K[2]^2\right )^2 \left (K[2]^4 \left (K[1]^2+2 K[2]^2\right )\right )^{3/2} \left (\sqrt {K[2]^4 \left (K[1]^2+2 K[2]^2\right )}-K[1] K[2]^2\right )^{2/3}}dK[1] K[2]^3+x^3 \sqrt [3]{2 \sqrt {K[2]^4 \left (x^2+2 K[2]^2\right )}-2 x K[2]^2} K[2]^2+x^2 \left (2 K[2]^4+\sqrt {K[2]^4 \left (x^2+2 K[2]^2\right )} \sqrt [3]{2 \sqrt {K[2]^4 \left (x^2+2 K[2]^2\right )}-2 x K[2]^2}\right )+x \left (2 K[2]^4 \sqrt [3]{2 \sqrt {K[2]^4 \left (x^2+2 K[2]^2\right )}-2 x K[2]^2}-\sqrt {K[2]^4 \left (x^2+2 K[2]^2\right )} \left (2 \sqrt {K[2]^4 \left (x^2+2 K[2]^2\right )}-2 x K[2]^2\right )^{2/3}\right )}{2 K[2]^3 \left (x^2+K[2]^2\right ) \left (x^2+2 K[2]^2\right )}dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {2 \left (2 \left (-i+\sqrt {3}\right ) K[3]+\left (i+\sqrt {3}\right ) \sqrt [3]{2 \sqrt {y(x)^4 \left (K[3]^2+2 y(x)^2\right )}-2 K[3] y(x)^2}\right ) y(x)^4+K[3]^2 \left (2 \left (-i+\sqrt {3}\right ) K[3]+\left (i+\sqrt {3}\right ) \sqrt [3]{2 \sqrt {y(x)^4 \left (K[3]^2+2 y(x)^2\right )}-2 K[3] y(x)^2}\right ) y(x)^2+\sqrt {y(x)^4 \left (K[3]^2+2 y(x)^2\right )} \sqrt [3]{2 \sqrt {y(x)^4 \left (K[3]^2+2 y(x)^2\right )}-2 K[3] y(x)^2} \left (\left (i+\sqrt {3}\right ) K[3]+2 i \sqrt [3]{2 \sqrt {y(x)^4 \left (K[3]^2+2 y(x)^2\right )}-2 K[3] y(x)^2}\right )}{2 \left (-i+\sqrt {3}\right ) (K[3]-i y(x)) (K[3]+i y(x)) y(x)^2 \left (K[3]^2+2 y(x)^2\right )}dK[3]+\int _1^{y(x)}-\frac {\left (\left (i+\sqrt {3}\right ) \sqrt [3]{2 \sqrt {K[4]^4 \left (x^2+2 K[4]^2\right )}-2 x K[4]^2} x^2+2 i \left (2 \sqrt {K[4]^4 \left (x^2+2 K[4]^2\right )}-2 x K[4]^2\right )^{2/3} x-2 \left (-i+\sqrt {3}\right ) \sqrt {K[4]^4 \left (x^2+2 K[4]^2\right )}\right ) K[4]^2+2 \left (-i+\sqrt {3}\right ) \left (x^2+K[4]^2\right ) \sqrt {K[4]^4 \left (x^2+2 K[4]^2\right )} \int _1^x-\frac {2 K[4]^5 \left (2 i 2^{2/3} K[4]^2 \sqrt [3]{\sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )}-K[3] K[4]^2} K[3]^5-i \sqrt [3]{2} \left (\left (3-3 i \sqrt {3}\right ) K[4]^4+2 \sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )} \sqrt [3]{2 \sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )}-2 K[3] K[4]^2}\right ) K[3]^4+\sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )} \left (\sqrt [3]{2} \left (i+\sqrt {3}\right ) K[4]^2+3 \left (-i+\sqrt {3}\right ) \left (\sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )}-K[3] K[4]^2\right )^{2/3}\right ) K[3]^3-i \sqrt [3]{2} K[4]^2 \left (\left (3-3 i \sqrt {3}\right ) K[4]^4+2 \sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )} \sqrt [3]{2 \sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )}-2 K[3] K[4]^2}\right ) K[3]^2+\left (-5 i 2^{2/3} \sqrt [3]{\sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )}-K[3] K[4]^2} K[4]^6+\sqrt [3]{2} \left (i+\sqrt {3}\right ) \sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )} K[4]^4+6 \left (-i+\sqrt {3}\right ) \sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )} \left (\sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )}-K[3] K[4]^2\right )^{2/3} K[4]^2\right ) K[3]+3 \sqrt [3]{2} K[4]^4 \left (\left (i+\sqrt {3}\right ) K[4]^4+i \sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )} \sqrt [3]{2 \sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )}-2 K[3] K[4]^2}\right )\right )}{3 \left (-i+\sqrt {3}\right ) \left (K[3]^2+K[4]^2\right )^2 \left (K[4]^4 \left (K[3]^2+2 K[4]^2\right )\right )^{3/2} \left (\sqrt {K[4]^4 \left (K[3]^2+2 K[4]^2\right )}-K[3] K[4]^2\right )^{2/3}}dK[3] K[4]+\left (i+\sqrt {3}\right ) x \sqrt {K[4]^4 \left (x^2+2 K[4]^2\right )} \sqrt [3]{2 \sqrt {K[4]^4 \left (x^2+2 K[4]^2\right )}-2 x K[4]^2}}{2 \left (-i+\sqrt {3}\right ) K[4] (K[4]-i x) (i x+K[4]) \sqrt {K[4]^4 \left (x^2+2 K[4]^2\right )}}dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {2 \left (2 \left (i+\sqrt {3}\right ) K[5]+\left (-i+\sqrt {3}\right ) \sqrt [3]{2 \sqrt {y(x)^4 \left (K[5]^2+2 y(x)^2\right )}-2 K[5] y(x)^2}\right ) y(x)^4+K[5]^2 \left (2 \left (i+\sqrt {3}\right ) K[5]+\left (-i+\sqrt {3}\right ) \sqrt [3]{2 \sqrt {y(x)^4 \left (K[5]^2+2 y(x)^2\right )}-2 K[5] y(x)^2}\right ) y(x)^2+\sqrt {y(x)^4 \left (K[5]^2+2 y(x)^2\right )} \sqrt [3]{2 \sqrt {y(x)^4 \left (K[5]^2+2 y(x)^2\right )}-2 K[5] y(x)^2} \left (\left (-i+\sqrt {3}\right ) K[5]-2 i \sqrt [3]{2 \sqrt {y(x)^4 \left (K[5]^2+2 y(x)^2\right )}-2 K[5] y(x)^2}\right )}{2 \left (i+\sqrt {3}\right ) (K[5]-i y(x)) (K[5]+i y(x)) y(x)^2 \left (K[5]^2+2 y(x)^2\right )}dK[5]+\int _1^{y(x)}\frac {2 \left (-\left (\left (-i+\sqrt {3}\right ) \sqrt [3]{2 \sqrt {K[6]^4 \left (x^2+2 K[6]^2\right )}-2 x K[6]^2} x^2\right )+2 i \left (2 \sqrt {K[6]^4 \left (x^2+2 K[6]^2\right )}-2 x K[6]^2\right )^{2/3} x+2 \left (i+\sqrt {3}\right ) \sqrt {K[6]^4 \left (x^2+2 K[6]^2\right )}\right ) K[6]^2-4 \left (i+\sqrt {3}\right ) \left (x^2+K[6]^2\right ) \sqrt {K[6]^4 \left (x^2+2 K[6]^2\right )} \int _1^x-\frac {2 K[6]^5 \left (-2 i 2^{2/3} K[6]^2 \sqrt [3]{\sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )}-K[5] K[6]^2} K[5]^5+i \sqrt [3]{2} \left (\left (3+3 i \sqrt {3}\right ) K[6]^4+2 \sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )} \sqrt [3]{2 \sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )}-2 K[5] K[6]^2}\right ) K[5]^4+\sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )} \left (\sqrt [3]{2} \left (-i+\sqrt {3}\right ) K[6]^2+3 \left (i+\sqrt {3}\right ) \left (\sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )}-K[5] K[6]^2\right )^{2/3}\right ) K[5]^3+i \sqrt [3]{2} K[6]^2 \left (\left (3+3 i \sqrt {3}\right ) K[6]^4+2 \sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )} \sqrt [3]{2 \sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )}-2 K[5] K[6]^2}\right ) K[5]^2+\left (5 i 2^{2/3} \sqrt [3]{\sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )}-K[5] K[6]^2} K[6]^6+\sqrt [3]{2} \left (-i+\sqrt {3}\right ) \sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )} K[6]^4+6 \left (i+\sqrt {3}\right ) \sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )} \left (\sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )}-K[5] K[6]^2\right )^{2/3} K[6]^2\right ) K[5]+3 \sqrt [3]{2} K[6]^4 \left (\left (-i+\sqrt {3}\right ) K[6]^4-i \sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )} \sqrt [3]{2 \sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )}-2 K[5] K[6]^2}\right )\right )}{3 \left (i+\sqrt {3}\right ) \left (K[5]^2+K[6]^2\right )^2 \left (K[6]^4 \left (K[5]^2+2 K[6]^2\right )\right )^{3/2} \left (\sqrt {K[6]^4 \left (K[5]^2+2 K[6]^2\right )}-K[5] K[6]^2\right )^{2/3}}dK[5] K[6]-2 \left (-i+\sqrt {3}\right ) x \sqrt {K[6]^4 \left (x^2+2 K[6]^2\right )} \sqrt [3]{2 \sqrt {K[6]^4 \left (x^2+2 K[6]^2\right )}-2 x K[6]^2}}{4 \left (i+\sqrt {3}\right ) K[6] (K[6]-i x) (i x+K[6]) \sqrt {K[6]^4 \left (x^2+2 K[6]^2\right )}}dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.23 (sec), leaf count = 896
\[\left [y \left (x \right ) = -\frac {i \sqrt {2}\, x}{2}, y \left (x \right ) = \frac {i \sqrt {2}\, x}{2}, y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {2 \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}} \textit {\_a}^{2}-2 \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {1}{3}} \textit {\_a}^{3}+\left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}}-\textit {\_a} \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {1}{3}}-\textit {\_a}^{2}}{\left (\textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{2}+1\right ) \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{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}\, \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}} \textit {\_a}^{2}+i \sqrt {3}\, \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}}+i \sqrt {3}\, \textit {\_a}^{2}-2 \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}} \textit {\_a}^{2}-4 \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {1}{3}} \textit {\_a}^{3}-\left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}}-2 \textit {\_a} \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {1}{3}}+\textit {\_a}^{2}}{\left (\textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{2}+1\right ) \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{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}\, \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}} \textit {\_a}^{2}+i \sqrt {3}\, \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}}+i \sqrt {3}\, \textit {\_a}^{2}+2 \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}} \textit {\_a}^{2}+4 \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {1}{3}} \textit {\_a}^{3}+\left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {1}{3}}-\textit {\_a}^{2}}{\left (\textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{2}+1\right ) \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {1}{3}}}d \textit {\_a} \right )+2 \textit {\_C1} \right ) x\right ]\] Mathematica raw input
DSolve[x + 3*y[x]*y'[x] + 2*y[x]*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][(y[x]^4*(4*K[1] - 2*(-2*K[1]*y[x]^2 + 2*Sqrt[y[x]^4*(K[1]^2 + 2*y[x]^2)])^(1/3)) + K[1]^2*y[x]^2*(2*K[1] - (-2*K[1]*y[x]^2 + 2*Sqrt[y[x]^4*(K[1]^2 + 2*y[x]^2)])^(1/3)) + Sqrt[y[x]^4*(K[1]^2 + 2*y[x]^2)]*(-2*K[1]*y[x]^2 + 2*Sqrt[y[x]^4*(K[1]^2 + 2*y[x]^2)])^(1/3)*(-K[1] + (-2*K[1]*y[x]^2 + 2*Sqrt[y[x]^4*(K[1]^2 + 2*y[x]^2)])^(1/3)))/(2*y[x]^2*(K[1]^2 + y[x]^2)*(K[1]^2 + 2*y[x]^2)), {K[1], 1, x}] + Inactive[Integrate][(4*K[2]^6 + x^3*K[2]^2*(-2*x*K[2]^2 + 2*Sqrt[K[2]^4*(x^2 + 2*K[2]^2)])^(1/3) + x^2*(2*K[2]^4 + Sqrt[K[2]^4*(x^2 + 2*K[2]^2)]*(-2*x*K[2]^2 + 2*Sqrt[K[2]^4*(x^2 + 2*K[2]^2)])^(1/3)) + x*(2*K[2]^4*(-2*x*K[2]^2 + 2*Sqrt[K[2]^4*(x^2 + 2*K[2]^2)])^(1/3) - Sqrt[K[2]^4*(x^2 + 2*K[2]^2)]*(-2*x*K[2]^2 + 2*Sqrt[K[2]^4*(x^2 + 2*K[2]^2)])^(2/3)) - 2*K[2]^3*(x^4 + 3*x^2*K[2]^2 + 2*K[2]^4)*Inactive[Integrate][(K[2]^5*(6*2^(1/3)*K[2]^8 - 2*2^(2/3)*K[1]^5*K[2]^2*(-(K[1]*K[2]^2) + Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)])^(1/3) - 3*2^(2/3)*K[2]^4*Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)]*(-(K[1]*K[2]^2) + Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)])^(1/3) + 2*K[1]^4*(-3*2^(1/3)*K[2]^4 + 2^(2/3)*Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)]*(-(K[1]*K[2]^2) + Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)])^(1/3)) + 2*K[1]^2*(-3*2^(1/3)*K[2]^6 + 2^(2/3)*K[2]^2*Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)]*(-(K[1]*K[2]^2) + Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)])^(1/3)) + 2*K[1]^3*Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)]*(2^(1/3)*K[2]^2 - 3*(-(K[1]*K[2]^2) + Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)])^(2/3)) + K[1]*(2*2^(1/3)*K[2]^4*Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)] + 5*2^(2/3)*K[2]^6*(-(K[1]*K[2]^2) + Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)])^(1/3) - 12*K[2]^2*Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)]*(-(K[1]*K[2]^2) + Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)])^(2/3))))/(3*(K[1]^2 + K[2]^2)^2*(K[2]^4*(K[1]^2 + 2*K[2]^2))^(3/2)*(-(K[1]*K[2]^2) + Sqrt[K[2]^4*(K[1]^2 + 2*K[2]^2)])^(2/3)), {K[1], 1, x}])/(2*K[2]^3*(x^2 + K[2]^2)*(x^2 + 2*K[2]^2)), {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(Sqrt[y[x]^4*(K[3]^2 + 2*y[x]^2)]*(-2*K[3]*y[x]^2 + 2*Sqrt[y[x]^4*(K[3]^2 + 2*y[x]^2)])^(1/3)*((I + Sqrt[3])*K[3] + (2*I)*(-2*K[3]*y[x]^2 + 2*Sqrt[y[x]^4*(K[3]^2 + 2*y[x]^2)])^(1/3)) + K[3]^2*y[x]^2*(2*(-I + Sqrt[3])*K[3] + (I + Sqrt[3])*(-2*K[3]*y[x]^2 + 2*Sqrt[y[x]^4*(K[3]^2 + 2*y[x]^2)])^(1/3)) + 2*y[x]^4*(2*(-I + Sqrt[3])*K[3] + (I + Sqrt[3])*(-2*K[3]*y[x]^2 + 2*Sqrt[y[x]^4*(K[3]^2 + 2*y[x]^2)])^(1/3)))/(2*(-I + Sqrt[3])*(K[3] - I*y[x])*(K[3] + I*y[x])*y[x]^2*(K[3]^2 + 2*y[x]^2)), {K[3], 1, x}] + Inactive[Integrate][-1/2*((I + Sqrt[3])*x*Sqrt[K[4]^4*(x^2 + 2*K[4]^2)]*(-2*x*K[4]^2 + 2*Sqrt[K[4]^4*(x^2 + 2*K[4]^2)])^(1/3) + K[4]^2*(-2*(-I + Sqrt[3])*Sqrt[K[4]^4*(x^2 + 2*K[4]^2)] + (I + Sqrt[3])*x^2*(-2*x*K[4]^2 + 2*Sqrt[K[4]^4*(x^2 + 2*K[4]^2)])^(1/3) + (2*I)*x*(-2*x*K[4]^2 + 2*Sqrt[K[4]^4*(x^2 + 2*K[4]^2)])^(2/3)) + 2*(-I + Sqrt[3])*K[4]*(x^2 + K[4]^2)*Sqrt[K[4]^4*(x^2 + 2*K[4]^2)]*Inactive[Integrate][(-2*K[4]^5*((2*I)*2^(2/3)*K[3]^5*K[4]^2*(-(K[3]*K[4]^2) + Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)])^(1/3) + K[3]^3*Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)]*(2^(1/3)*(I + Sqrt[3])*K[4]^2 + 3*(-I + Sqrt[3])*(-(K[3]*K[4]^2) + Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)])^(2/3)) + K[3]*(2^(1/3)*(I + Sqrt[3])*K[4]^4*Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)] - (5*I)*2^(2/3)*K[4]^6*(-(K[3]*K[4]^2) + Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)])^(1/3) + 6*(-I + Sqrt[3])*K[4]^2*Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)]*(-(K[3]*K[4]^2) + Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)])^(2/3)) + 3*2^(1/3)*K[4]^4*((I + Sqrt[3])*K[4]^4 + I*Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)]*(-2*K[3]*K[4]^2 + 2*Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)])^(1/3)) - I*2^(1/3)*K[3]^4*((3 - (3*I)*Sqrt[3])*K[4]^4 + 2*Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)]*(-2*K[3]*K[4]^2 + 2*Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)])^(1/3)) - I*2^(1/3)*K[3]^2*K[4]^2*((3 - (3*I)*Sqrt[3])*K[4]^4 + 2*Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)]*(-2*K[3]*K[4]^2 + 2*Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)])^(1/3))))/(3*(-I + Sqrt[3])*(K[3]^2 + K[4]^2)^2*(K[4]^4*(K[3]^2 + 2*K[4]^2))^(3/2)*(-(K[3]*K[4]^2) + Sqrt[K[4]^4*(K[3]^2 + 2*K[4]^2)])^(2/3)), {K[3], 1, x}])/((-I + Sqrt[3])*K[4]*((-I)*x + K[4])*(I*x + K[4])*Sqrt[K[4]^4*(x^2 + 2*K[4]^2)]), {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(Sqrt[y[x]^4*(K[5]^2 + 2*y[x]^2)]*(-2*K[5]*y[x]^2 + 2*Sqrt[y[x]^4*(K[5]^2 + 2*y[x]^2)])^(1/3)*((-I + Sqrt[3])*K[5] - (2*I)*(-2*K[5]*y[x]^2 + 2*Sqrt[y[x]^4*(K[5]^2 + 2*y[x]^2)])^(1/3)) + K[5]^2*y[x]^2*(2*(I + Sqrt[3])*K[5] + (-I + Sqrt[3])*(-2*K[5]*y[x]^2 + 2*Sqrt[y[x]^4*(K[5]^2 + 2*y[x]^2)])^(1/3)) + 2*y[x]^4*(2*(I + Sqrt[3])*K[5] + (-I + Sqrt[3])*(-2*K[5]*y[x]^2 + 2*Sqrt[y[x]^4*(K[5]^2 + 2*y[x]^2)])^(1/3)))/(2*(I + Sqrt[3])*(K[5] - I*y[x])*(K[5] + I*y[x])*y[x]^2*(K[5]^2 + 2*y[x]^2)), {K[5], 1, x}] + Inactive[Integrate][(-2*(-I + Sqrt[3])*x*Sqrt[K[6]^4*(x^2 + 2*K[6]^2)]*(-2*x*K[6]^2 + 2*Sqrt[K[6]^4*(x^2 + 2*K[6]^2)])^(1/3) + 2*K[6]^2*(2*(I + Sqrt[3])*Sqrt[K[6]^4*(x^2 + 2*K[6]^2)] - (-I + Sqrt[3])*x^2*(-2*x*K[6]^2 + 2*Sqrt[K[6]^4*(x^2 + 2*K[6]^2)])^(1/3) + (2*I)*x*(-2*x*K[6]^2 + 2*Sqrt[K[6]^4*(x^2 + 2*K[6]^2)])^(2/3)) - 4*(I + Sqrt[3])*K[6]*(x^2 + K[6]^2)*Sqrt[K[6]^4*(x^2 + 2*K[6]^2)]*Inactive[Integrate][(-2*K[6]^5*((-2*I)*2^(2/3)*K[5]^5*K[6]^2*(-(K[5]*K[6]^2) + Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)])^(1/3) + K[5]^3*Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)]*(2^(1/3)*(-I + Sqrt[3])*K[6]^2 + 3*(I + Sqrt[3])*(-(K[5]*K[6]^2) + Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)])^(2/3)) + K[5]*(2^(1/3)*(-I + Sqrt[3])*K[6]^4*Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)] + (5*I)*2^(2/3)*K[6]^6*(-(K[5]*K[6]^2) + Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)])^(1/3) + 6*(I + Sqrt[3])*K[6]^2*Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)]*(-(K[5]*K[6]^2) + Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)])^(2/3)) + 3*2^(1/3)*K[6]^4*((-I + Sqrt[3])*K[6]^4 - I*Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)]*(-2*K[5]*K[6]^2 + 2*Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)])^(1/3)) + I*2^(1/3)*K[5]^4*((3 + (3*I)*Sqrt[3])*K[6]^4 + 2*Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)]*(-2*K[5]*K[6]^2 + 2*Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)])^(1/3)) + I*2^(1/3)*K[5]^2*K[6]^2*((3 + (3*I)*Sqrt[3])*K[6]^4 + 2*Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)]*(-2*K[5]*K[6]^2 + 2*Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)])^(1/3))))/(3*(I + Sqrt[3])*(K[5]^2 + K[6]^2)^2*(K[6]^4*(K[5]^2 + 2*K[6]^2))^(3/2)*(-(K[5]*K[6]^2) + Sqrt[K[6]^4*(K[5]^2 + 2*K[6]^2)])^(2/3)), {K[5], 1, x}])/(4*(I + Sqrt[3])*K[6]*((-I)*x + K[6])*(I*x + K[6])*Sqrt[K[6]^4*(x^2 + 2*K[6]^2)]), {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(2*y(x)*diff(y(x),x)^3+3*y(x)*diff(y(x),x)+x = 0, y(x))
Maple raw output
[y(x) = -1/2*I*2^(1/2)*x, y(x) = 1/2*I*2^(1/2)*x, y(x) = RootOf(-ln(x)+Intat((2*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(2/3)*_a^2-2*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(1/3)*_a^3+(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(2/3)-_a*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(1/3)-_a^2)/(_a^2+1)/(2*_a^2+1)/(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(1/3),_a = _Z)+_C1)*x, y(x) = RootOf(-2*ln(x)+Intat((2*I*3^(1/2)*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(2/3)*_a^2+I*3^(1/2)*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(2/3)+I*3^(1/2)*_a^2-2*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(2/3)*_a^2-4*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(1/3)*_a^3-(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(2/3)-2*_a*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(1/3)+_a^2)/(_a^2+1)/(2*_a^2+1)/(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(1/3),_a = _Z)+2*_C1)*x, y(x) = RootOf(-2*ln(x)-Intat((2*I*3^(1/2)*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(2/3)*_a^2+I*3^(1/2)*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(2/3)+I*3^(1/2)*_a^2+2*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(2/3)*_a^2+4*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(1/3)*_a^3+(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(2/3)+2*_a*(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(1/3)-_a^2)/(_a^2+1)/(2*_a^2+1)/(-_a*(-2*_a^4+(2*_a^2+1)^(3/2)-3*_a^2-1)/(2*_a^2+1)^(5/2))^(1/3),_a = _Z)+2*_C1)*x]