\[ a+x y'(x)^2+y(x) y'(x)=0 \] ✓ Mathematica : cpu = 0.388556 (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 = 0.056 (sec), leaf count = 146
\[-\frac {c_{1} \left (\frac {-y \relax (x )+\sqrt {y \relax (x )^{2}-4 a x}}{x}\right )^{\frac {3}{2}} x^{2}}{\left (-y \relax (x )+\sqrt {y \relax (x )^{2}-4 a x}\right )^{2}}+x +\frac {4 a \,x^{2}}{3 \left (-y \relax (x )+\sqrt {y \relax (x )^{2}-4 a x}\right )^{2}} = 0\]