#### 2.412   ODE No. 412

$a+x y'(x)^2+y(x) y'(x)=0$ Mathematica : cpu = 0.39402 (sec), leaf count = 3229

$\left \{\left \{y(x)\to -\frac {1}{2} \sqrt {-\frac {16 a x}{9}+\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}+\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}} x}+\frac {e^{3 c_1}}{1296 a^2 x^2}}-\frac {1}{2} \sqrt {-\frac {\frac {128}{27} e^{\frac {3 c_1}{2}}+\frac {e^{\frac {9 c_1}{2}}}{5832 a^3 x^3}}{4 \sqrt {-\frac {16 a x}{9}+\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}+\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}} x}+\frac {e^{3 c_1}}{1296 a^2 x^2}}}-\frac {32 a x}{9}-\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}-\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a x \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}+\frac {e^{3 c_1}}{648 a^2 x^2}}+\frac {e^{\frac {3 c_1}{2}}}{72 a x}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {-\frac {16 a x}{9}+\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}+\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}} x}+\frac {e^{3 c_1}}{1296 a^2 x^2}}+\frac {1}{2} \sqrt {-\frac {\frac {128}{27} e^{\frac {3 c_1}{2}}+\frac {e^{\frac {9 c_1}{2}}}{5832 a^3 x^3}}{4 \sqrt {-\frac {16 a x}{9}+\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}+\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}} x}+\frac {e^{3 c_1}}{1296 a^2 x^2}}}-\frac {32 a x}{9}-\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}-\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a x \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}+\frac {e^{3 c_1}}{648 a^2 x^2}}+\frac {e^{\frac {3 c_1}{2}}}{72 a x}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {-\frac {16 a x}{9}+\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}+\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}} x}+\frac {e^{3 c_1}}{1296 a^2 x^2}}-\frac {1}{2} \sqrt {\frac {\frac {128}{27} e^{\frac {3 c_1}{2}}+\frac {e^{\frac {9 c_1}{2}}}{5832 a^3 x^3}}{4 \sqrt {-\frac {16 a x}{9}+\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}+\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}} x}+\frac {e^{3 c_1}}{1296 a^2 x^2}}}-\frac {32 a x}{9}-\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}-\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a x \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}+\frac {e^{3 c_1}}{648 a^2 x^2}}+\frac {e^{\frac {3 c_1}{2}}}{72 a x}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {-\frac {16 a x}{9}+\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}+\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}} x}+\frac {e^{3 c_1}}{1296 a^2 x^2}}+\frac {1}{2} \sqrt {\frac {\frac {128}{27} e^{\frac {3 c_1}{2}}+\frac {e^{\frac {9 c_1}{2}}}{5832 a^3 x^3}}{4 \sqrt {-\frac {16 a x}{9}+\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}+\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}} x}+\frac {e^{3 c_1}}{1296 a^2 x^2}}}-\frac {32 a x}{9}-\frac {\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}{18\ 2^{2/3} a x}-\frac {4\ 2^{2/3} \left (128 a^4 x^4+a e^{3 c_1} x\right )}{9 a x \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {-1073741824 a^9 e^{3 c_1} x^9+3145728 a^6 e^{6 c_1} x^6-3072 a^3 e^{9 c_1} x^3+e^{12 c_1}}}}+\frac {e^{3 c_1}}{648 a^2 x^2}}+\frac {e^{\frac {3 c_1}{2}}}{72 a x}\right \}\right \}$ Maple : cpu = 0.195 (sec), leaf count = 146

$\left \{ -{{\it \_C1}\,{x}^{2} \left ( {\frac {1}{x} \left ( -y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-4\,ax} \right ) } \right ) ^{{\frac {3}{2}}} \left ( -y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-4\,ax} \right ) ^{-2}}+x+{\frac {4\,a{x}^{2}}{3} \left ( -y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-4\,ax} \right ) ^{-2}}=0,{{\it \_C1}\,{x}^{2} \left ( {\frac {1}{x} \left ( -2\,y \left ( x \right ) -2\,\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-4\,ax} \right ) } \right ) ^{{\frac {3}{2}}} \left ( y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-4\,ax} \right ) ^{-2}}+x+{\frac {4\,a{x}^{2}}{3} \left ( y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-4\,ax} \right ) ^{-2}}=0 \right \}$