#### 2.672   ODE No. 672

$y'(x)=\frac {x^2 \left (\sqrt {4 y(x)^3-9 x^4}+3 x\right )}{y(x)^2}$ Mathematica : cpu = 1.90354 (sec), leaf count = 4512

$\text {Solve}\left [\int _1^x\left (-\frac {24 \sqrt {4 y(x)^3-9 K[1]^4} y(x)^9}{\left (9 K[1]^4-4 y(x)^3\right ) \left (4 y(x)^9-729\right )}+\frac {16 K[1]^2 \sqrt {4 y(x)^3-9 K[1]^4} y(x)^9}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}+\frac {24 \sqrt {4 y(x)^3-9 K[1]^4} y(x)^9}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}+\frac {108 K[1] \sqrt {4 y(x)^3-9 K[1]^4} y(x)^6}{\left (9 K[1]^4-4 y(x)^3\right ) \left (4 y(x)^9-729\right )}-\frac {48 K[1]^3 \sqrt {4 y(x)^3-9 K[1]^4} y(x)^6}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}-\frac {108 K[1] \sqrt {4 y(x)^3-9 K[1]^4} y(x)^6}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}-\frac {96 \sqrt {4 y(x)^3-9 K[1]^4} y(x)^6}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}-\frac {486 K[1]^2 \sqrt {4 y(x)^3-9 K[1]^4} y(x)^3}{\left (9 K[1]^4-4 y(x)^3\right ) \left (4 y(x)^9-729\right )}+\frac {216 K[1]^4 \sqrt {4 y(x)^3-9 K[1]^4} y(x)^3}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}+\frac {486 K[1]^2 \sqrt {4 y(x)^3-9 K[1]^4} y(x)^3}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}+\frac {432 K[1] \sqrt {4 y(x)^3-9 K[1]^4} y(x)^3}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}+\frac {4 \left (2 K[1]^5+3 K[1]^3+4 K[1]^2\right )}{4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16}+\frac {2187 K[1]^3 \sqrt {4 y(x)^3-9 K[1]^4}}{\left (9 K[1]^4-4 y(x)^3\right ) \left (4 y(x)^9-729\right )}-\frac {972 K[1]^5 \sqrt {4 y(x)^3-9 K[1]^4}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}-\frac {2187 K[1]^3 \sqrt {4 y(x)^3-9 K[1]^4}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}-\frac {4860 K[1]^2 \sqrt {4 y(x)^3-9 K[1]^4}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 y(x)^3+16\right ) \left (4 y(x)^9-729\right )}\right )dK[1]+\int _1^{y(x)}\left (-\frac {2 K[2]^2}{\left (x^3+2\right ) \sqrt {4 K[2]^3-9 x^4}}+\frac {2 \sqrt {4 K[2]^3-9 x^4} K[2]^2}{\left (x^3+2\right ) \left (-4 x^6-9 x^4-16 x^3+4 K[2]^3-16\right )}+\frac {4 K[2]^2}{-4 x^6-9 x^4-16 x^3+4 K[2]^3-16}-\int _1^x\left (\frac {864 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{17}}{\left (9 K[1]^4-4 K[2]^3\right ) \left (4 K[2]^9-729\right )^2}-\frac {576 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{17}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}-\frac {864 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{17}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}-\frac {3888 K[1] \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{14}}{\left (9 K[1]^4-4 K[2]^3\right ) \left (4 K[2]^9-729\right )^2}+\frac {1728 K[1]^3 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{14}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}+\frac {3888 K[1] \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{14}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}+\frac {3456 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{14}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}-\frac {288 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{11}}{\left (9 K[1]^4-4 K[2]^3\right )^2 \left (4 K[2]^9-729\right )}+\frac {192 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{11}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}+\frac {288 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{11}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}-\frac {144 K[2]^{11}}{\left (9 K[1]^4-4 K[2]^3\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}+\frac {96 K[1]^2 K[2]^{11}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}+\frac {144 K[2]^{11}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}+\frac {17496 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{11}}{\left (9 K[1]^4-4 K[2]^3\right ) \left (4 K[2]^9-729\right )^2}-\frac {7776 K[1]^4 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{11}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}-\frac {17496 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{11}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}-\frac {15552 K[1] \sqrt {4 K[2]^3-9 K[1]^4} K[2]^{11}}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}-\frac {216 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (9 K[1]^4-4 K[2]^3\right ) \left (4 K[2]^9-729\right )}+\frac {144 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )}+\frac {216 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )}+\frac {1296 K[1] \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (9 K[1]^4-4 K[2]^3\right )^2 \left (4 K[2]^9-729\right )}-\frac {576 K[1]^3 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}-\frac {1296 K[1] \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}-\frac {1152 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}+\frac {648 K[1] K[2]^8}{\left (9 K[1]^4-4 K[2]^3\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}-\frac {288 K[1]^3 K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}-\frac {648 K[1] K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}-\frac {576 K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}-\frac {78732 K[1]^3 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (9 K[1]^4-4 K[2]^3\right ) \left (4 K[2]^9-729\right )^2}+\frac {34992 K[1]^5 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}+\frac {78732 K[1]^3 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}+\frac {174960 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^8}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )^2}+\frac {648 K[1] \sqrt {4 K[2]^3-9 K[1]^4} K[2]^5}{\left (9 K[1]^4-4 K[2]^3\right ) \left (4 K[2]^9-729\right )}-\frac {288 K[1]^3 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^5}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )}-\frac {648 K[1] \sqrt {4 K[2]^3-9 K[1]^4} K[2]^5}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )}-\frac {576 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^5}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )}-\frac {5832 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^5}{\left (9 K[1]^4-4 K[2]^3\right )^2 \left (4 K[2]^9-729\right )}+\frac {2592 K[1]^4 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^5}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}+\frac {5832 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^5}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}+\frac {5184 K[1] \sqrt {4 K[2]^3-9 K[1]^4} K[2]^5}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}-\frac {2916 K[1]^2 K[2]^5}{\left (9 K[1]^4-4 K[2]^3\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}+\frac {1296 K[1]^4 K[2]^5}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}+\frac {2916 K[1]^2 K[2]^5}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}+\frac {2592 K[1] K[2]^5}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}-\frac {1458 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^2}{\left (9 K[1]^4-4 K[2]^3\right ) \left (4 K[2]^9-729\right )}+\frac {648 K[1]^4 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^2}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )}+\frac {1458 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^2}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )}+\frac {1296 K[1] \sqrt {4 K[2]^3-9 K[1]^4} K[2]^2}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \left (4 K[2]^9-729\right )}+\frac {26244 K[1]^3 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^2}{\left (9 K[1]^4-4 K[2]^3\right )^2 \left (4 K[2]^9-729\right )}-\frac {11664 K[1]^5 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^2}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}-\frac {26244 K[1]^3 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^2}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}-\frac {58320 K[1]^2 \sqrt {4 K[2]^3-9 K[1]^4} K[2]^2}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2 \left (4 K[2]^9-729\right )}+\frac {13122 K[1]^3 K[2]^2}{\left (9 K[1]^4-4 K[2]^3\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}-\frac {5832 K[1]^5 K[2]^2}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}-\frac {13122 K[1]^3 K[2]^2}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}-\frac {29160 K[1]^2 K[2]^2}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right ) \sqrt {4 K[2]^3-9 K[1]^4} \left (4 K[2]^9-729\right )}+\frac {48 \left (2 K[1]^5+3 K[1]^3+4 K[1]^2\right ) K[2]^2}{\left (4 K[1]^6+9 K[1]^4+16 K[1]^3-4 K[2]^3+16\right )^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ]$ Maple : cpu = 0.327 (sec), leaf count = 36

$\left \{ \int _{{\it \_b}}^{y \left ( x \right ) }\!{{{\it \_a}}^{2}{\frac {1}{\sqrt {-9\,{x}^{4}+4\,{{\it \_a}}^{3}}}}}\,{\rm d}{\it \_a}-{\frac {{x}^{3}}{3}}-{\it \_C1}=0 \right \}$