#### 2.288   ODE No. 288

$\left (-3 x^2 y(x)+6 y(x)^2+1\right ) y'(x)-3 x y(x)^2+x=0$ Mathematica : cpu = 0.20842 (sec), leaf count = 534

$\left \{\left \{y(x)\to \frac {x^2}{4}-\frac {\sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8+207 x^4-54 c_1 x^6+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}{4\ 3^{2/3}}+\frac {6-\frac {9 x^4}{4}}{3 \sqrt [3]{3} \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8+207 x^4-54 c_1 x^6+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}\right \},\left \{y(x)\to \frac {x^2}{4}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8+207 x^4-54 c_1 x^6+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}{8\ 3^{2/3}}-\frac {\left (1+i \sqrt {3}\right ) \left (6-\frac {9 x^4}{4}\right )}{6 \sqrt [3]{3} \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8+207 x^4-54 c_1 x^6+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}\right \},\left \{y(x)\to \frac {x^2}{4}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8+207 x^4-54 c_1 x^6+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}{8\ 3^{2/3}}-\frac {\left (1-i \sqrt {3}\right ) \left (6-\frac {9 x^4}{4}\right )}{6 \sqrt [3]{3} \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8+207 x^4-54 c_1 x^6+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}\right \}\right \}$ Maple : cpu = 0.03 (sec), leaf count = 579

$\left \{ y \left ( x \right ) =-{\frac {1}{24} \left ( -6\,{x}^{2}\sqrt [3]{-324\,{x}^{2}-432\,{\it \_C1}+27\,{x}^{6}+12\,\sqrt {-81\,{x}^{8}-162\,{\it \_C1}\,{x}^{6}+621\,{x}^{4}+1944\,{\it \_C1}\,{x}^{2}+1296\,{{\it \_C1}}^{2}+96}}+ \left ( -9\,i{x}^{4}+i \left ( -324\,{x}^{2}-432\,{\it \_C1}+27\,{x}^{6}+12\,\sqrt {-81\,{x}^{8}-162\,{\it \_C1}\,{x}^{6}+621\,{x}^{4}+1944\,{\it \_C1}\,{x}^{2}+1296\,{{\it \_C1}}^{2}+96} \right ) ^{{\frac {2}{3}}}+24\,i \right ) \sqrt {3}+9\,{x}^{4}+ \left ( -324\,{x}^{2}-432\,{\it \_C1}+27\,{x}^{6}+12\,\sqrt {-81\,{x}^{8}-162\,{\it \_C1}\,{x}^{6}+621\,{x}^{4}+1944\,{\it \_C1}\,{x}^{2}+1296\,{{\it \_C1}}^{2}+96} \right ) ^{{\frac {2}{3}}}-24 \right ) {\frac {1}{\sqrt [3]{-324\,{x}^{2}-432\,{\it \_C1}+27\,{x}^{6}+12\,\sqrt {-81\,{x}^{8}-162\,{\it \_C1}\,{x}^{6}+621\,{x}^{4}+1944\,{\it \_C1}\,{x}^{2}+1296\,{{\it \_C1}}^{2}+96}}}}},y \left ( x \right ) ={\frac {1}{24} \left ( 6\,{x}^{2}\sqrt [3]{-324\,{x}^{2}-432\,{\it \_C1}+27\,{x}^{6}+12\,\sqrt {-81\,{x}^{8}-162\,{\it \_C1}\,{x}^{6}+621\,{x}^{4}+1944\,{\it \_C1}\,{x}^{2}+1296\,{{\it \_C1}}^{2}+96}}+ \left ( -9\,i{x}^{4}+i \left ( -324\,{x}^{2}-432\,{\it \_C1}+27\,{x}^{6}+12\,\sqrt {-81\,{x}^{8}-162\,{\it \_C1}\,{x}^{6}+621\,{x}^{4}+1944\,{\it \_C1}\,{x}^{2}+1296\,{{\it \_C1}}^{2}+96} \right ) ^{{\frac {2}{3}}}+24\,i \right ) \sqrt {3}-9\,{x}^{4}- \left ( -324\,{x}^{2}-432\,{\it \_C1}+27\,{x}^{6}+12\,\sqrt {-81\,{x}^{8}-162\,{\it \_C1}\,{x}^{6}+621\,{x}^{4}+1944\,{\it \_C1}\,{x}^{2}+1296\,{{\it \_C1}}^{2}+96} \right ) ^{{\frac {2}{3}}}+24 \right ) {\frac {1}{\sqrt [3]{-324\,{x}^{2}-432\,{\it \_C1}+27\,{x}^{6}+12\,\sqrt {-81\,{x}^{8}-162\,{\it \_C1}\,{x}^{6}+621\,{x}^{4}+1944\,{\it \_C1}\,{x}^{2}+1296\,{{\it \_C1}}^{2}+96}}}}},y \left ( x \right ) ={\frac {1}{12}\sqrt [3]{-324\,{x}^{2}-432\,{\it \_C1}+27\,{x}^{6}+12\,\sqrt {-81\,{x}^{8}-162\,{\it \_C1}\,{x}^{6}+621\,{x}^{4}+1944\,{\it \_C1}\,{x}^{2}+1296\,{{\it \_C1}}^{2}+96}}}+{\frac {3\,{x}^{4}-8}{4}{\frac {1}{\sqrt [3]{-324\,{x}^{2}-432\,{\it \_C1}+27\,{x}^{6}+12\,\sqrt {-81\,{x}^{8}-162\,{\it \_C1}\,{x}^{6}+621\,{x}^{4}+1944\,{\it \_C1}\,{x}^{2}+1296\,{{\it \_C1}}^{2}+96}}}}}+{\frac {{x}^{2}}{4}} \right \}$