ODE
\[ x y''(x)+2 y'(x)+x y(x)^5=0 \] ODE Classification
[_Emden, [_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 97.87 (sec), leaf count = 11691
\[\left \{\text {Solve}\left [\frac {2 \sqrt {3} \sqrt {x} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) y(x) \sqrt {\frac {\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ] \sqrt {\frac {\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} \sqrt {-4 x^3 y(x)^6+3 x y(x)^2+12 c_1}}=c_2+\int _1^x-\frac {12 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) K[1]^3 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^3+3 c_1\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} y(x)^9+4 \sqrt {3} K[1]^{5/2} \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]{}^2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]{}^2+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2\right ) K[1]^3+12 c_1\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \sqrt {-4 K[1]^3 y(x)^6+3 K[1] y(x)^2+12 c_1} y(x)^8-6 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) K[1]^4 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (-4 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]{}^2 K[1]^2+4 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^2+3\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} y(x)^7-4 \sqrt {3} K[1]^{11/2} \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]+2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \sqrt {-4 K[1]^3 y(x)^6+3 K[1] y(x)^2+12 c_1} y(x)^6+9 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) K[1] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (4 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^2+1\right ) K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (K[1] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]-12 c_1\right ) K[1]^2+3 c_1\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} y(x)^5-3 \sqrt {3} \sqrt {K[1]} \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]{}^2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]{}^2+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2\right ) K[1]^3+12 c_1\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \sqrt {-4 K[1]^3 y(x)^6+3 K[1] y(x)^2+12 c_1} y(x)^4+72 c_1 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^3+3 c_1\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} y(x)^3-\frac {3 \sqrt {3} \left (4 c_1 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\left (4 c_1-K[1] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]{}^2-2 K[1] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]+4 c_1 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2\right ) K[1]^3+4 c_1 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 K[1]^3+\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]{}^2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^3+12 c_1\right )\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \sqrt {-4 K[1]^3 y(x)^6+3 K[1] y(x)^2+12 c_1} y(x)^2}{\sqrt {K[1]}}-9 c_1 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) K[1] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (-4 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]{}^2 K[1]^2+4 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] K[1]^2+3\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} y(x)+12 \sqrt {3} c_1 K[1]^{5/2} \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]+2 \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \sqrt {-4 K[1]^3 y(x)^6+3 K[1] y(x)^2+12 c_1}}{\sqrt {3} K[1]^{7/2} \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) \sqrt {\frac {\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \left (y(x)^2-\text {Root}\left [4 K[1]^3 \text {$\#$1}^3-3 K[1] \text {$\#$1}-12 c_1\& ,3\right ]\right ) \left (-4 K[1]^3 y(x)^6+3 K[1] y(x)^2+12 c_1\right ){}^{3/2}}dK[1],y(x)\right ],\text {Solve}\left [\frac {2 \sqrt {3} \sqrt {x} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) y(x) \sqrt {\frac {\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \left (y(x)^2-\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]\right )}{\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ] \sqrt {\frac {\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 x^3 \text {$\#$1}^3-3 x \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} \sqrt {-4 x^3 y(x)^6+3 x y(x)^2+12 c_1}}=c_2+\int _1^x-\frac {-12 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) K[2]^3 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^3+3 c_1\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} y(x)^9+4 \sqrt {3} K[2]^{5/2} \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]{}^2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]{}^2+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2\right ) K[2]^3+12 c_1\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \sqrt {-4 K[2]^3 y(x)^6+3 K[2] y(x)^2+12 c_1} y(x)^8+6 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) K[2]^4 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (-4 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]{}^2 K[2]^2+4 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^2+3\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} y(x)^7-4 \sqrt {3} K[2]^{11/2} \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]+2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \sqrt {-4 K[2]^3 y(x)^6+3 K[2] y(x)^2+12 c_1} y(x)^6-9 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) K[2] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (4 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^2+1\right ) K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (K[2] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]-12 c_1\right ) K[2]^2+3 c_1\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} y(x)^5-3 \sqrt {3} \sqrt {K[2]} \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]{}^2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]{}^2+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2\right ) K[2]^3+12 c_1\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \sqrt {-4 K[2]^3 y(x)^6+3 K[2] y(x)^2+12 c_1} y(x)^4-72 c_1 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^3+3 c_1\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} y(x)^3-\frac {3 \sqrt {3} \left (4 c_1 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]{}^2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\left (4 c_1-K[2] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]{}^2-2 K[2] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]+4 c_1 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2\right ) K[2]^3+4 c_1 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 K[2]^3+\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]{}^2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^3+12 c_1\right )\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \sqrt {-4 K[2]^3 y(x)^6+3 K[2] y(x)^2+12 c_1} y(x)^2}{\sqrt {K[2]}}+9 c_1 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}}\right )|\frac {\left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]\right ) \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right )}\right ) K[2] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (-4 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]{}^2 K[2]^2+4 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] K[2]^2+3\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) y(x)^2 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]{}^2 \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right ){}^2}} y(x)+12 \sqrt {3} c_1 K[2]^{5/2} \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]+2 \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \sqrt {-4 K[2]^3 y(x)^6+3 K[2] y(x)^2+12 c_1}}{\sqrt {3} K[2]^{7/2} \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) \sqrt {\frac {\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ]-y(x)^2\right )}{\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,2\right ] \left (\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,1\right ]-y(x)^2\right )}} \left (y(x)^2-\text {Root}\left [4 K[2]^3 \text {$\#$1}^3-3 K[2] \text {$\#$1}-12 c_1\& ,3\right ]\right ) \left (-4 K[2]^3 y(x)^6+3 K[2] y(x)^2+12 c_1\right ){}^{3/2}}dK[2],y(x)\right ]\right \}\]
Maple ✓
cpu = 1.007 (sec), leaf count = 73
\[\left [y \left (x \right ) = \frac {\RootOf \left (\ln \left (x \right )-6 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {-12 \textit {\_f}^{6}+9 \textit {\_f}^{2}+9 \textit {\_C1}}}d \textit {\_f} \right )+2 \textit {\_C2} \right )}{\sqrt {x}}, y \left (x \right ) = \frac {\RootOf \left (\ln \left (x \right )+6 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {-12 \textit {\_f}^{6}+9 \textit {\_f}^{2}+9 \textit {\_C1}}}d \textit {\_f} \right )+2 \textit {\_C2} \right )}{\sqrt {x}}\right ]\] Mathematica raw input
DSolve[x*y[x]^5 + 2*y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
{Solve[(2*Sqrt[3]*Sqrt[x]*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 2])*Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3])/(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 2]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3]))]*y[x]*Sqrt[(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 2]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - y[x]^2))]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3]*Sqrt[(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - y[x]^2)^2)]*Sqrt[12*C[1] + 3*x*y[x]^2 - 4*x^3*y[x]^6]) == C[2] + Inactive[Integrate][-((-9*C[1]*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2])*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]))]*K[1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(3 - 4*K[1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]^2 + 4*K[1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2)^2)] + 72*C[1]*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2])*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]))]*(3*C[1] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2)*y[x]^3*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2)^2)] + 9*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2])*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]))]*K[1]*(3*C[1] + K[1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(-12*C[1] + K[1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]) + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(1 + 4*K[1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]))*y[x]^5*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2)^2)] - 6*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2])*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]))]*K[1]^4*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(3 - 4*K[1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]^2 + 4*K[1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^7*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2)^2)] + 12*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2])*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]))]*K[1]^3*(3*C[1] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2)*y[x]^9*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2)^2)] + 12*Sqrt[3]*C[1]*K[1]^(5/2)*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2] + 2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]*Sqrt[12*C[1] + 3*K[1]*y[x]^2 - 4*K[1]^3*y[x]^6] - (3*Sqrt[3]*(4*C[1]*K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] + 4*C[1]*(12*C[1] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2) + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(4*C[1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2 - 2*K[1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2 + Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]^2*(4*C[1] - K[1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])))*y[x]^2*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]*Sqrt[12*C[1] + 3*K[1]*y[x]^2 - 4*K[1]^3*y[x]^6])/Sqrt[K[1]] - 3*Sqrt[3]*Sqrt[K[1]]*(12*C[1] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2 + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]^2 + Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2))*y[x]^4*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]*Sqrt[12*C[1] + 3*K[1]*y[x]^2 - 4*K[1]^3*y[x]^6] - 4*Sqrt[3]*K[1]^(11/2)*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2] + 2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*y[x]^6*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]*Sqrt[12*C[1] + 3*K[1]*y[x]^2 - 4*K[1]^3*y[x]^6] + 4*Sqrt[3]*K[1]^(5/2)*(12*C[1] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]^2*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2 + K[1]^3*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]^2 + Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3]^2))*y[x]^8*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]*Sqrt[12*C[1] + 3*K[1]*y[x]^2 - 4*K[1]^3*y[x]^6])/(Sqrt[3]*K[1]^(7/2)*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3])*Sqrt[(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 1] - y[x]^2))]*(-Root[-12*C[1] - 3*K[1]*#1 + 4*K[1]^3*#1^3 & , 3] + y[x]^2)*(12*C[1] + 3*K[1]*y[x]^2 - 4*K[1]^3*y[x]^6)^(3/2))), {K[1], 1, x}], y[x]], Solve[(2*Sqrt[3]*Sqrt[x]*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 2])*Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3])/(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 2]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3]))]*y[x]*Sqrt[(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 2]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - y[x]^2))]*(-Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3] + y[x]^2))/(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3]*Sqrt[(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1]*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*x*#1 + 4*x^3*#1^3 & , 1] - y[x]^2)^2)]*Sqrt[12*C[1] + 3*x*y[x]^2 - 4*x^3*y[x]^6]) == C[2] + Inactive[Integrate][-((9*C[1]*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2])*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]))]*K[2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(3 - 4*K[2]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]^2 + 4*K[2]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2)^2)] - 72*C[1]*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2])*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]))]*(3*C[1] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2)*y[x]^3*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2)^2)] - 9*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2])*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]))]*K[2]*(3*C[1] + K[2]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(-12*C[1] + K[2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]) + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(1 + 4*K[2]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]))*y[x]^5*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2)^2)] + 6*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2])*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]))]*K[2]^4*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(3 - 4*K[2]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]^2 + 4*K[2]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^7*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2)^2)] - 12*EllipticF[ArcSin[Sqrt[((Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^2)/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]], ((Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2])*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]))]*K[2]^3*(3*C[1] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2)*y[x]^9*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^2*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2)^2)] + 12*Sqrt[3]*C[1]*K[2]^(5/2)*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2] + 2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]*Sqrt[12*C[1] + 3*K[2]*y[x]^2 - 4*K[2]^3*y[x]^6] - (3*Sqrt[3]*(4*C[1]*K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] + 4*C[1]*(12*C[1] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2) + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(4*C[1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2 - 2*K[2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2 + Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]^2*(4*C[1] - K[2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])))*y[x]^2*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]*Sqrt[12*C[1] + 3*K[2]*y[x]^2 - 4*K[2]^3*y[x]^6])/Sqrt[K[2]] - 3*Sqrt[3]*Sqrt[K[2]]*(12*C[1] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2 + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]^2 + Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2))*y[x]^4*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]*Sqrt[12*C[1] + 3*K[2]*y[x]^2 - 4*K[2]^3*y[x]^6] - 4*Sqrt[3]*K[2]^(11/2)*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2] + 2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*y[x]^6*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]*Sqrt[12*C[1] + 3*K[2]*y[x]^2 - 4*K[2]^3*y[x]^6] + 4*Sqrt[3]*K[2]^(5/2)*(12*C[1] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]^2*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2 + K[2]^3*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]^2 + Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3]^2))*y[x]^8*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]*Sqrt[12*C[1] + 3*K[2]*y[x]^2 - 4*K[2]^3*y[x]^6])/(Sqrt[3]*K[2]^(7/2)*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3])*Sqrt[(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2] - y[x]^2))/(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 2]*(Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 1] - y[x]^2))]*(-Root[-12*C[1] - 3*K[2]*#1 + 4*K[2]^3*#1^3 & , 3] + y[x]^2)*(12*C[1] + 3*K[2]*y[x]^2 - 4*K[2]^3*y[x]^6)^(3/2))), {K[2], 1, x}], y[x]]}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+2*diff(y(x),x)+x*y(x)^5 = 0, y(x))
Maple raw output
[y(x) = RootOf(ln(x)-6*Intat(1/(-12*_f^6+9*_f^2+9*_C1)^(1/2),_f = _Z)+2*_C2)/x^(1/2), y(x) = RootOf(ln(x)+6*Intat(1/(-12*_f^6+9*_f^2+9*_C1)^(1/2),_f = _Z)+2*_C2)/x^(1/2)]