\[ {}y^{\prime } = \frac {y}{x \left (-1+y x +x y^{3}+y^{4} x \right )} \]

\[ \frac {1}{x y}+\frac {y^{3}}{3}+\frac {y^{2}}{2}+\ln \left (y\right ) = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y \left (y x +1\right )}{x \left (-y x -1+y^{4} x^{3}\right )} \]

\[ \frac {3 x y+2}{6 y^{3} x^{3}}+y = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y \left (x +y\right )}{x \left (x +y+y^{3}+y^{4}\right )} \]

\[ -\ln \left (x \right )-\frac {x}{y}+\frac {y^{3}}{3}+\frac {y^{2}}{2}+\ln \left (y\right ) = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y \left (x^{3}+x^{2} y+y^{2}\right )}{x^{2} \left (-1+x \right ) \left (x +y\right )} \]

\[ \frac {\ln \left (y^{2}+x y+x^{2}\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 y+x \right ) \sqrt {3}}{3 x}\right )}{3}-\ln \left (y\right ) = -\ln \left (x -1\right )+\ln \left (x \right )+c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y \left (x -y\right )}{x \left (x -y-y^{3}-y^{4}\right )} \]

\[ -\ln \left (x \right )+\frac {x}{y}+\frac {y^{3}}{3}+\frac {y^{2}}{2}+\ln \left (y\right ) = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {b^{3}+y^{2} b^{3}+2 y b^{2} a x +x^{2} b \,a^{2}+y^{3} b^{3}+3 y^{2} b^{2} a x +3 y b \,a^{2} x^{2}+a^{3} x^{3}}{b^{3}} \]

\[ \int _{}^{y-\frac {a x}{b}-\frac {1}{3}}\frac {27 b}{27 \textit {\_a}^{3} b -9 \textit {\_a} b +27 a +29 b}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = \frac {\alpha ^{3}+y^{2} \alpha ^{3}+2 y \alpha ^{2} \beta x +\alpha \,\beta ^{2} x^{2}+y^{3} \alpha ^{3}+3 y^{2} \alpha ^{2} \beta x +3 y \alpha \,\beta ^{2} x^{2}+\beta ^{3} x^{3}}{\alpha ^{3}} \]

\[ \int _{}^{y-\frac {\beta x}{\alpha }-\frac {1}{3}}\frac {27 \alpha }{27 \textit {\_a}^{3} \alpha -9 \textit {\_a} \alpha +29 \alpha +27 \beta }d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = \frac {14 y x +12+2 x +x^{3} y^{3}+6 y^{2} x^{2}}{x^{2} \left (y x +2+x \right )} \]

\[ -x -\frac {2}{y^{2} \left (x y+2\right )^{2}}+\frac {2 y+2}{y^{2} \left (x y+2\right )}-\frac {1}{2 y^{2}}-\frac {1}{y} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {a^{3}+y^{2} a^{3}+2 y a^{2} b x +a \,b^{2} x^{2}+y^{3} a^{3}+3 y^{2} b x \,a^{2}+3 y a \,b^{2} x^{2}+b^{3} x^{3}}{a^{3}} \]

\[ \int _{}^{y-\frac {b x}{a}-\frac {1}{3}}\frac {27 a}{27 \textit {\_a}^{3} a -9 \textit {\_a} a +29 a +27 b}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = \frac {-\sin \left (2 y\right )+x \cos \left (2 y\right )+\cos \left (2 y\right ) x^{3}+\cos \left (2 y\right ) x^{4}+x +x^{3}+x^{4}}{2 x} \]

\[ -\frac {4 x \left (\left (x^{4}+\frac {5}{4} x^{3}+\frac {5}{2} x \right ) \cos \left (2 y\right )+x^{4}+\frac {5 x^{3}}{4}+\frac {5 x}{2}-5 \sin \left (2 y\right )\right )}{20 \cos \left (2 y\right )+20} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y \left ({\mathrm e}^{-\frac {x^{2}}{2}} x y+{\mathrm e}^{-\frac {x^{2}}{4}} x +2 y^{2} {\mathrm e}^{-\frac {3 x^{2}}{4}}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2 y \,{\mathrm e}^{-\frac {x^{2}}{4}}+2} \]

\[ \frac {{\mathrm e}^{\frac {x^{2}}{4}} \left (2 y+{\mathrm e}^{\frac {x^{2}}{4}}\right )}{4 y^{2}} = -\frac {x}{2}+c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {2 x}{3}+1+y^{2}+\frac {2 x^{2} y}{3}+\frac {x^{4}}{9}+y^{3}+y^{2} x^{2}+\frac {x^{4} y}{3}+\frac {x^{6}}{27} \]

\[ \int _{}^{y-\frac {x^{2}}{3}-\frac {1}{3}}\frac {1}{\textit {\_a}^{3}+\frac {29}{27}-\frac {1}{3} \textit {\_a}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = 2 x +1+y^{2}-2 x^{2} y+x^{4}+y^{3}-3 y^{2} x^{2}+3 x^{4} y-x^{6} \]

\[ \int _{}^{y+x^{2}-\frac {1}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = \frac {-30 x^{3} y+12 x^{6}+70 x^{\frac {7}{2}}-30 x^{3}-25 y \sqrt {x}+50 x -25 \sqrt {x}-25}{5 \left (-5 y+2 x^{3}+10 \sqrt {x}-5\right ) x} \]

\[ 20 x^{\frac {7}{2}}+\left (-50 y-50\right ) \sqrt {x}-25 \ln \left (x \right )+2 x^{6}+\left (-10 y-10\right ) x^{3}+50 x +\frac {25 y^{2}}{2}+25 y = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {\left (-256 a \,x^{2}+512+512 y^{2}+128 y a \,x^{4}+8 a^{2} x^{8}+512 y^{3}+192 x^{4} a y^{2}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512} \]

\[ \int _{}^{y-\frac {\frac {3}{8} a \,x^{5}+x}{3 x}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {x^{2}}{2}-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = -\frac {y^{2} \left (x^{2} y-2 x -2 y x +y\right )}{2 \left (-2+y x -2 y\right ) x} \]

\[ \frac {-1+\left (x -2\right ) y}{y^{2}} = \frac {x^{2}}{4}-x +\frac {\ln \left (x \right )}{2}+c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-2 y x +2 x^{3}-2 x -y^{3}+3 y^{2} x^{2}-3 x^{4} y+x^{6}}{-y+x^{2}-1} \]

\[ \frac {-2 x^{2}+2 y+1}{2 \left (-y+x^{2}\right )^{2}} = -x +c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-18 y x -6 x^{3}-18 x +27 y^{3}+27 y^{2} x^{2}+9 x^{4} y+x^{6}}{27 y+9 x^{2}+27} \]

\[ \frac {-2 x^{2}-6 y-3}{\left (x^{2}+3 y\right )^{2}} = \frac {2 x}{3}+c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {\left (-108 x^{\frac {3}{2}}-216-216 y^{2}+72 x^{3} y-6 x^{6}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{216} \]

\[ \int _{}^{y-\frac {-\frac {x^{\frac {7}{2}}}{2}+\sqrt {x}}{3 \sqrt {x}}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {2 x^{\frac {3}{2}}}{3}-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {a^{2} x y+a +x \,a^{2}+y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 a x y+1}{a^{2} x^{2} \left (a x y+1+x a \right )} \]

\[ -x -\frac {1}{2 y^{2} \left (y a x +1\right )^{2}}+\frac {y+1}{y^{2} \left (y a x +1\right )}-\frac {1}{y}-\frac {1}{2 y^{2}} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {6 x^{2} y-2 x +1-5 x^{3} y^{2}-2 y x +x^{4} y^{3}}{x^{2} \left (x^{2} y-x +1\right )} \]

\[ \frac {-2 x^{2} y+2 x -1}{x^{2} \left (x y-1\right )^{2}} = -\frac {2}{x}+c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {6 x +x^{3}+x^{3} y^{2}+4 x^{2} y+x^{3} y^{3}+6 y^{2} x^{2}+12 y x +8}{x^{3}} \]

\[ \int _{}^{y-\frac {1}{3}-\frac {2}{x}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = \frac {32 x^{5}+64 x^{6}+64 y^{2} x^{6}+32 x^{4} y+4 x^{2}+64 x^{6} y^{3}+48 x^{4} y^{2}+12 x^{2} y+1}{64 x^{8}} \]

\[ \int _{}^{y-\frac {\left (\frac {1}{x^{2}}+\frac {3}{4 x^{4}}\right ) x^{2}}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} +\frac {1}{x}-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {\left (y-a \ln \left (y\right ) x +x^{2}\right ) y}{\left (-y \ln \left (y\right )-y \ln \left (x \right )-y+x a \right ) x} \]

\[ \frac {x^{2}}{2}-a \ln \left (y\right ) x +y \ln \left (x \right )+y \ln \left (y\right ) = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {x \,a^{2}+a^{3} x^{3}+a^{3} x^{3} y^{2}+2 a^{2} x^{2} y+x a +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 a x y+1}{a^{3} x^{3}} \]

\[ \int _{}^{y-\frac {1}{3}-\frac {1}{x a}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = \frac {-2 x -y+1+y^{2} x^{2}+2 x^{3} y+x^{4}+x^{3} y^{3}+3 x^{4} y^{2}+3 x^{5} y+x^{6}}{x} \]

\[ \int _{}^{x}\frac {-783-729 \textit {\_a}^{3} \left (y-\frac {3 x^{3}+x}{3 x^{2}}\right )^{3}+243 \left (\textit {\_a} +3\right ) \left (y-\frac {3 x^{3}+x}{3 x^{2}}\right )}{27 \textit {\_a}^{3} \left (y-\frac {3 x^{3}+x}{3 x^{2}}\right )^{3}-9 \textit {\_a} \left (y-\frac {3 x^{3}+x}{3 x^{2}}\right )+29}d \textit {\_a} = c_{3} \] Verified OK.

\[ {}y^{\prime } = -\left (-\frac {\ln \left (y\right )}{x}+\frac {\cos \left (x \right ) \ln \left (y\right )}{\sin \left (x \right )}-f_{1} \left (x \right )\right ) y \]

\[ \int _{}^{x}-\frac {\ln \left (y\right ) \sin \left (\textit {\_a} \right )-\cos \left (\textit {\_a} \right ) \ln \left (y\right ) \textit {\_a} +f_{1} \left (\textit {\_a} \right ) \textit {\_a} \sin \left (\textit {\_a} \right )}{\textit {\_a}^{2}}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = -\left (-\frac {\ln \left (y\right )}{x}+\frac {\ln \left (y\right )}{x \ln \left (x \right )}-f_{1} \left (x \right )\right ) y \]

\[ \int _{}^{x}-\frac {f_{1} \left (\textit {\_a} \right ) \textit {\_a} \ln \left (\textit {\_a} \right )+\ln \left (\textit {\_a} \right ) \ln \left (y\right )-\ln \left (y\right )}{\textit {\_a}^{2}}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y^{2}}{y^{2}+y^{\frac {3}{2}}+\sqrt {y}\, x^{2}-2 y^{\frac {3}{2}} x +y^{\frac {5}{2}}+x^{3}-3 x^{2} y+3 x y^{2}-y^{3}} \]

\[ \frac {\ln \left (y\right )}{2} = \int _{}^{\frac {x -y}{\sqrt {y}}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}-\textit {\_a} +2}d \textit {\_a} +c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {\left (-\frac {\ln \left (y\right )^{2}}{2 x}-f_{1} \left (x \right )\right ) y}{\ln \left (y\right )} \]

\[ \int _{}^{x}-\frac {2 \textit {\_a} f_{1} \left (\textit {\_a} \right )+\ln \left (y\right )^{2}}{\textit {\_a}^{2}}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-8 x^{2} y^{3}+16 x y^{2}+16 x y^{3}-8+12 y x -6 y^{2} x^{2}+x^{3} y^{3}}{16 \left (-2+y x -2 y\right ) x} \]

\[ \frac {-16-8 x^{2} y+\left (16 y+16\right ) x}{x^{2} \left (y x -2\right )^{2}} = \frac {\ln \left (x \right )}{2}+\frac {4}{x}-\frac {4}{x^{2}}+c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {\left (-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8-8 y^{2}+8 x^{2} {\mathrm e}^{-x^{2}} y-2 x^{4} {\mathrm e}^{-2 x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{8} \]

\[ \int _{}^{y-\frac {-\frac {3 \,{\mathrm e}^{-x^{2}} x^{3}}{2}+x}{3 x}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {x^{2}}{2}-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {-3 x^{2} y-2 x^{3}-2 x -x y^{2}-y+x^{3} y^{3}+3 x^{4} y^{2}+3 x^{5} y+x^{6}}{x \left (y x +x^{2}+1\right )} \]

\[ \ln \left (x +y\right )-\frac {\ln \left (1+2 x^{5}+\left (4 y+1\right ) x^{4}+\left (2 y^{2}+2 y\right ) x^{3}+\left (y^{2}+2\right ) x^{2}+2 x y\right )}{2} = -\ln \left (x \right )+c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {-x^{2}-y x -x^{3}-x y^{2}+2 y x^{2} \ln \left (x \right )-x^{3} \ln \left (x \right )^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x^{2}} \]

\[ 27 \left (\int _{}^{\frac {y-\frac {\left (-\frac {3 \ln \left (x \right )}{x}+\frac {1}{x}\right ) x^{2}}{3}}{x}}\frac {1}{27 \textit {\_a}^{3}-9 \textit {\_a} +29}d \textit {\_a} \right ) = x +c_{4} \] Verified OK.

\[ {}y^{\prime } = \frac {x}{2}+1+y^{2}+\frac {x^{2} y}{4}-y x -\frac {x^{4}}{8}+\frac {x^{3}}{8}+\frac {x^{2}}{4}+y^{3}-\frac {3 y^{2} x^{2}}{4}-\frac {3 x y^{2}}{2}+\frac {3 x^{4} y}{16}+\frac {3 x^{3} y}{4}-\frac {x^{6}}{64}-\frac {3 x^{5}}{32} \]

\[ \int _{}^{y-\frac {1}{3}+\frac {x^{2}}{4}+\frac {x}{2}}\frac {1}{\frac {31}{54}+\textit {\_a}^{3}-\frac {1}{3} \textit {\_a}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = -\frac {x}{2}+1+y^{2}+\frac {7 x^{2} y}{2}-2 y x +\frac {13 x^{4}}{16}-\frac {3 x^{3}}{2}+x^{2}+y^{3}+\frac {3 y^{2} x^{2}}{4}-3 x y^{2}+\frac {3 x^{4} y}{16}-\frac {3 x^{3} y}{2}+\frac {x^{6}}{64}-\frac {3 x^{5}}{16} \]

\[ \int _{}^{y-\frac {1}{3}-\frac {x^{2}}{4}+x}\frac {1}{-\frac {1}{3} \textit {\_a} +\frac {2}{27}+\textit {\_a}^{3}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = -\frac {x}{4}+1+y^{2}+\frac {7 x^{2} y}{16}-\frac {y x}{2}+\frac {5 x^{4}}{128}-\frac {5 x^{3}}{64}+\frac {x^{2}}{16}+y^{3}+\frac {3 y^{2} x^{2}}{8}-\frac {3 x y^{2}}{4}+\frac {3 x^{4} y}{64}-\frac {3 x^{3} y}{16}+\frac {x^{6}}{512}-\frac {3 x^{5}}{256} \]

\[ \int _{}^{y-\frac {1}{3}-\frac {x^{2}}{8}+\frac {x}{4}}\frac {1}{-\frac {1}{3} \textit {\_a} +\frac {89}{108}+\textit {\_a}^{3}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = \frac {-x^{2}+x +1+y^{2}+5 x^{2} y-2 y x +4 x^{4}-3 x^{3}+y^{3}+3 y^{2} x^{2}-3 x y^{2}+3 x^{4} y-6 x^{3} y+x^{6}-3 x^{5}}{x} \]

\[ \int _{}^{y-\frac {\left (3 x -3+\frac {1}{x}\right ) x}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {-32 y x +16 x^{3}+16 x^{2}-32 x -64 y^{3}+48 y^{2} x^{2}+96 x y^{2}-12 x^{4} y-48 x^{3} y-48 x^{2} y+x^{6}+6 x^{5}+12 x^{4}}{-64 y+16 x^{2}+32 x -64} \]

\[ x = \frac {4 \ln \left (-\frac {x^{2}}{4}-\frac {x}{2}+y-1\right )}{5}-\frac {2 \ln \left (2 \left (-\frac {x^{2}}{4}-\frac {x}{2}+y\right )^{2}-\frac {x^{2}}{2}-x +2 y+1\right )}{5}+\frac {2 \arctan \left (\frac {x^{2}}{2}-2 y+x -1\right )}{5}+c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-32 y x -72 x^{3}+32 x^{2}-32 x +64 y^{3}+48 y^{2} x^{2}-192 x y^{2}+12 x^{4} y-96 x^{3} y+192 x^{2} y+x^{6}-12 x^{5}+48 x^{4}}{64 y+16 x^{2}-64 x +64} \]

\[ x = \int _{}^{y}\frac {\frac {1}{4} x^{2}-x +\textit {\_a} +1}{\left (\frac {1}{4} x^{2}-x +\textit {\_a} \right )^{3}-\frac {x^{2}}{4}+x -\textit {\_a} -1}d \textit {\_a} +c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-128 y x -24 x^{3}+32 x^{2}-128 x +512 y^{3}+192 y^{2} x^{2}-384 x y^{2}+24 x^{4} y-96 x^{3} y+96 x^{2} y+x^{6}-6 x^{5}+12 x^{4}}{512 y+64 x^{2}-128 x +512} \]

\[ x = \int _{}^{y}\frac {\frac {1}{2} x^{2}-x +4 \textit {\_a} +4}{4 \left (\frac {1}{8} x^{2}-\frac {1}{4} x +\textit {\_a} \right )^{3}-\frac {x^{2}}{8}+\frac {x}{4}-\textit {\_a} -1}d \textit {\_a} +c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-32 a x y-8 a^{2} x^{3}-16 a b \,x^{2}-32 x a +64 y^{3}+48 x^{2} a y^{2}+96 y^{2} b x +12 y a^{2} x^{4}+48 y a \,x^{3} b +48 y b^{2} x^{2}+a^{3} x^{6}+6 a^{2} x^{5} b +12 a \,x^{4} b^{2}+8 b^{3} x^{3}}{64 y+16 a \,x^{2}+32 b x +64} \]

\[ -\frac {b x}{2} = \int _{}^{y}-\frac {b \left (\frac {1}{4} a \,x^{2}+\frac {1}{2} b x +\textit {\_a} +1\right )}{2 \left (\frac {1}{4} a \,x^{2}+\frac {1}{2} b x +\textit {\_a} \right )^{3}+\left (\frac {1}{4} a \,x^{2}+\frac {1}{2} b x +\textit {\_a} \right ) b +b}d \textit {\_a} +c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-32 y x -8 x^{3}-16 a \,x^{2}-32 x +64 y^{3}+48 y^{2} x^{2}+96 a x y^{2}+12 x^{4} y+48 y a \,x^{3}+48 a^{2} x^{2} y+x^{6}+6 x^{5} a +12 a^{2} x^{4}+8 a^{3} x^{3}}{64 y+16 x^{2}+32 x a +64} \]

\[ -\frac {x}{2} = \int _{}^{y}-\frac {\frac {1}{4} x^{2}+\frac {1}{2} a x +\textit {\_a} +1}{2 \left (\frac {1}{4} x^{2}+\frac {1}{2} a x +\textit {\_a} \right )^{3}+\left (\frac {1}{4} x^{2}+\frac {1}{2} a x +\textit {\_a} \right ) a +a}d \textit {\_a} +c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {x^{2} y+x^{4}+2 x^{3}-3 x^{2}+y x +x +y^{3}+3 y^{2} x^{2}-3 x y^{2}+3 x^{4} y-6 x^{3} y+x^{6}-3 x^{5}}{x \left (y+x^{2}-x +1\right )} \]

\[ \frac {-2 x^{2}+2 x -2 y-1}{\left (x^{2}-x +y\right )^{2}} = 2 \ln \left (x \right )+c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {x a}{2}+1+y^{2}+\frac {a \,x^{2} y}{2}+b x y+\frac {a^{2} x^{4}}{16}+\frac {a b \,x^{3}}{4}+\frac {b^{2} x^{2}}{4}+y^{3}+\frac {3 x^{2} a y^{2}}{4}+\frac {3 y^{2} b x}{2}+\frac {3 y a^{2} x^{4}}{16}+\frac {3 y a \,x^{3} b}{4}+\frac {3 y b^{2} x^{2}}{4}+\frac {a^{3} x^{6}}{64}+\frac {3 a^{2} x^{5} b}{32}+\frac {3 a \,x^{4} b^{2}}{16}+\frac {b^{3} x^{3}}{8} \]

\[ \int _{}^{y-\frac {1}{3}-\frac {a \,x^{2}}{4}-\frac {b x}{2}}\frac {1}{\frac {1}{2} b +\frac {29}{27}-\frac {1}{3} \textit {\_a} +\textit {\_a}^{3}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = -\frac {x}{2}+1+y^{2}+\frac {x^{2} y}{2}+a x y+\frac {x^{4}}{16}+\frac {a \,x^{3}}{4}+\frac {x^{2} a^{2}}{4}+y^{3}+\frac {3 y^{2} x^{2}}{4}+\frac {3 a x y^{2}}{2}+\frac {3 x^{4} y}{16}+\frac {3 y a \,x^{3}}{4}+\frac {3 a^{2} x^{2} y}{4}+\frac {x^{6}}{64}+\frac {3 x^{5} a}{32}+\frac {3 a^{2} x^{4}}{16}+\frac {a^{3} x^{3}}{8} \]

\[ \int _{}^{y-\frac {1}{3}-\frac {x^{2}}{4}-\frac {a x}{2}}\frac {1}{\frac {1}{2} a +\frac {29}{27}+\textit {\_a}^{3}-\frac {1}{3} \textit {\_a}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = \frac {150 x^{3}+125 \sqrt {x}+125+125 y^{2}-100 x^{3} y-500 y \sqrt {x}+20 x^{6}+200 x^{\frac {7}{2}}+500 x +125 y^{3}-150 x^{3} y^{2}-750 y^{2} \sqrt {x}+60 y x^{6}+600 y x^{\frac {7}{2}}+1500 y x -8 x^{9}-120 x^{\frac {13}{2}}-600 x^{4}-1000 x^{\frac {3}{2}}}{125 x} \]

\[ \int _{}^{y-\frac {\left (-\frac {6 x^{2}}{5}-\frac {6}{\sqrt {x}}+\frac {1}{x}\right ) x}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {2 x +4 y \ln \left (1+2 x \right ) x +6 y^{2} \ln \left (1+2 x \right ) x +6 y \ln \left (1+2 x \right )^{2} x +2 \ln \left (1+2 x \right )^{3} x +2 x y^{3}+2 \ln \left (1+2 x \right )^{2} x +2 x y^{2}-1+3 y^{2} \ln \left (1+2 x \right )+3 y \ln \left (1+2 x \right )^{2}+y^{2}+y^{3}+2 y \ln \left (1+2 x \right )+\ln \left (1+2 x \right )^{2}+\ln \left (1+2 x \right )^{3}}{1+2 x} \]

\[ \int _{}^{y-\frac {2 \ln \left (2 x +1\right ) x}{2 x +1}-\frac {\ln \left (2 x +1\right )}{2 x +1}-\frac {2 x}{3 \left (2 x +1\right )}-\frac {1}{3 \left (2 x +1\right )}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y^{\prime } = \frac {-y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{3} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x} \]

\[ \int _{}^{x}-\frac {\sin \left (\frac {y}{\textit {\_a}}\right ) {\mathrm e}^{-\textit {\_a}^{2}} \left (\sin \left (\frac {y}{\textit {\_a}}\right ) \textit {\_a}^{3}+\cos \left (\frac {y}{\textit {\_a}}\right ) y\right )}{\textit {\_a}^{2}}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{2} \sin \left (\frac {y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x} \]

\[ \frac {{\mathrm e}^{-2 x} \sin \left (\frac {y}{x}\right )^{2}}{2} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {-4 x \cos \left (x \right )+4 x^{2} \sin \left (x \right )+4 x +4+4 y^{2}+8 y \cos \left (x \right ) x -8 y x +2 x^{2} \cos \left (2 x \right )+6 x^{2}-8 x^{2} \cos \left (x \right )+4 y^{3}+12 y^{2} \cos \left (x \right ) x -12 x y^{2}+6 y x^{2} \cos \left (2 x \right )+18 x^{2} y-24 y \cos \left (x \right ) x^{2}+x^{3} \cos \left (3 x \right )+15 x^{3} \cos \left (x \right )-6 x^{3} \cos \left (2 x \right )-10 x^{3}}{4 x} \]

\[ \int _{}^{y-\frac {\left (3 \cos \left (x \right )-3+\frac {1}{x}\right ) x}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {-y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x +2 \sin \left (\frac {y}{x}\right ) x^{3} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{4} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x} \]

\[ \int _{}^{x}-\frac {\sin \left (\frac {y}{\textit {\_a}}\right ) {\mathrm e}^{-\textit {\_a}^{2}-\frac {2}{3} \textit {\_a}^{3}} \left (\left (\textit {\_a}^{4}+\textit {\_a}^{3}+\textit {\_a} \right ) \sin \left (\frac {y}{\textit {\_a}}\right )+\cos \left (\frac {y}{\textit {\_a}}\right ) y\right )}{\textit {\_a}^{4}}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = -\frac {x \left (-513-1296 y^{2} x^{2}-432 x -456 x^{6}-972 x^{4} y^{2}-576 x^{5}-96 x^{8}+64 x^{9}-864 x^{4}-144 x^{7}-216 y^{3}-540 y^{2}-1134 x^{2}-378 y-756 x^{3}-594 x^{2} y-648 x^{2} y^{3}+432 x^{3} y^{2}+720 x^{3} y+1008 x^{5} y-288 y x^{6}-288 y x^{8}+288 y x^{7}+864 y^{2} x^{5}+432 y^{2} x^{7}-648 x^{4} y^{3}-216 x^{4} y-216 y^{2} x^{6}-216 x^{6} y^{3}\right )}{216 \left (x^{2}+1\right )^{4}} \]

\[ \int _{}^{y-\frac {-\frac {2 x^{8}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}-\frac {4 x^{6}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {9 x^{5}}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}-\frac {2 x^{4}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {6 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {5 x}{2 \left (x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1\right )}}{3 \left (\frac {x^{7}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {3 x^{5}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {3 x^{3}}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}+\frac {x}{x^{8}+4 x^{6}+6 x^{4}+4 x^{2}+1}\right )}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {\ln \left (x^{2}+1\right )}{2}-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {-\sin \left (\frac {y}{x}\right ) y x -y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x +y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x +y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{4} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x \left (1+x \right )} \]

\[ \int _{}^{x}-\frac {\sin \left (\frac {y}{\textit {\_a}}\right ) {\mathrm e}^{-\textit {\_a} \left (\textit {\_a} -2\right )} \left (y \left (\textit {\_a} +1\right ) \cos \left (\frac {y}{\textit {\_a}}\right )+\sin \left (\frac {y}{\textit {\_a}}\right ) \textit {\_a}^{4}\right )}{\textit {\_a}^{2} \left (\textit {\_a} +1\right )^{3}}d \textit {\_a} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x +y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x +y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )-\sin \left (\frac {y}{x}\right ) y x -y \sin \left (\frac {y}{x}\right )+2 \sin \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x}{2 \cos \left (\frac {y}{x}\right ) \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right ) \left (1+x \right )} \]

\[ \frac {\left (2 x +1\right ) \sin \left (\frac {y}{x}\right )^{2}-\cos \left (\frac {y}{x}\right )^{2} x^{2}}{2 x^{2}} = c_{1} \] Verified OK.

\[ {}y^{\prime } = \frac {\left (y x +1\right )^{3}}{x^{5}} \]

\[ \frac {\ln \left (y-\frac {1}{x}-1\right )}{3}-\frac {\ln \left (\left (y-\frac {1}{x}\right )^{2}+y-\frac {1}{x}+1\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 y-\frac {2}{x}+1\right ) \sqrt {3}}{3}\right )}{3}+\frac {1}{x}-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {y \left (y^{2}+y x +x^{2}+x \right )}{x^{2}} \]

\[ 27 \left (\int _{}^{\frac {y-\frac {x}{3}}{x}}\frac {1}{27 \textit {\_a}^{3}+18 \textit {\_a} -7}d \textit {\_a} \right ) = x +c_{4} \] Verified OK.

\[ {}y^{\prime } = \frac {y^{3}-3 x y^{2}+3 x^{2} y-x^{3}+x}{x} \]

\[ -\frac {1}{2 \left (y+x \right )^{2}}-\ln \left (x \right )-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {y^{3}-3 x y^{2}+3 x^{2} y-x^{3}+x^{2}}{\left (-1+x \right ) \left (1+x \right )} \]

\[ -\frac {\ln \left (\left (y+x \right )^{2}-y-x +1\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 y+2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\ln \left (y+x +1\right )}{3}+\operatorname {arctanh}\left (x \right )-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime } = \frac {\left (y x +1\right ) \left (y^{2} x^{2}+x^{2} y+2 y x +1+x +x^{2}\right )}{x^{5}} \]

\[ \int _{}^{y-\frac {\left (\frac {1}{x^{2}}+\frac {3}{x^{3}}\right ) x^{2}}{3}}\frac {1}{\textit {\_a}^{3}+\frac {2}{3} \textit {\_a} -\frac {34}{27}}d \textit {\_a} +\frac {1}{x}-c_{2} = 0 \] Verified OK.

\[ {}y^{\prime \prime }-y^{2} = 0 \]

\[ \int _{}^{y}\frac {3}{\sqrt {6 \textit {\_a}^{3}+18 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {3}{\sqrt {6 \textit {\_a}^{3}+18 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }-6 y^{2} = 0 \]

\[ \int _{}^{y}\frac {1}{\sqrt {4 \textit {\_a}^{3}+2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {4 \textit {\_a}^{3}+2 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }-6 y^{2}+4 y = 0 \]

\[ \int _{}^{y}\frac {1}{\sqrt {4 \textit {\_a}^{3}-4 \textit {\_a}^{2}+2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {4 \textit {\_a}^{3}-4 \textit {\_a}^{2}+2 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }-a y^{3} = 0 \]

\[ \int _{}^{y}\frac {2}{\sqrt {2 \textit {\_a}^{4} a +8 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {2}{\sqrt {2 \textit {\_a}^{4} a +8 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }+d +b y^{2}+c y+a y^{3} = 0 \]

\[ \int _{}^{y}\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }+6 a^{10} y^{11}-y = 0 \]

\[ \int _{}^{y}\frac {1}{\sqrt {-\textit {\_a}^{12} a^{10}+\textit {\_a}^{2}+2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {-\textit {\_a}^{12} a^{10}+\textit {\_a}^{2}+2 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }+a \sin \left (y\right ) = 0 \]

\[ \frac {2 \sqrt {\frac {a \cos \left (y\right )+c_{1}}{c_{1} +a}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {\sqrt {2}\, \sqrt {a}}{\sqrt {c_{1} +a}}\right )}{\sqrt {2 a \cos \left (y\right )+2 c_{1}}} = x +c_{2} \] Verified OK.

\[ -\frac {2 \sqrt {\frac {a \cos \left (y\right )+c_{1}}{c_{1} +a}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {\sqrt {2}\, \sqrt {a}}{\sqrt {c_{1} +a}}\right )}{\sqrt {2 a \cos \left (y\right )+2 c_{1}}} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }+y y^{\prime }-y^{3} = 0 \]

\[ \int _{}^{y}\frac {2 \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}}{\textit {\_a}^{4}-\textit {\_a}^{2} \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}+\left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}\frac {4 \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}}{-i \textit {\_a}^{4} \sqrt {3}+i \sqrt {3}\, \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {2}{3}}-\textit {\_a}^{4}-2 \textit {\_a}^{2} \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}-\left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ \int _{}^{y}-\frac {4 \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}}{-i \textit {\_a}^{4} \sqrt {3}+i \sqrt {3}\, \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {2}{3}}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}+\left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{4} \] Verified OK.

\[ {}y^{\prime \prime }+a y y^{\prime }+b y^{3} = 0 \]

\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (b \,\textit {\_a}^{4}+a \,\textit {\_a}^{2} \textit {\_Z} +2 \textit {\_Z}^{2}-{\mathrm e}^{\operatorname {RootOf}\left ({\tanh \left (\frac {\sqrt {a^{2}-8 b}\, \left (4 c_{1} -\textit {\_Z} \right )}{2 a}\right )}^{2} \textit {\_a}^{4} a^{2}-8 {\tanh \left (\frac {\sqrt {a^{2}-8 b}\, \left (4 c_{1} -\textit {\_Z} \right )}{2 a}\right )}^{2} \textit {\_a}^{4} b -a^{2} \textit {\_a}^{4}+8 b \,\textit {\_a}^{4}-8 \,{\mathrm e}^{\textit {\_Z}}\right )}\right )}d \textit {\_a} = x +c_{2} \] Warning, solution could not be verified

\[ {}y^{\prime \prime }+a {y^{\prime }}^{2}+b y = 0 \]

\[ \int _{}^{y}\frac {2 \,{\mathrm e}^{2 a \textit {\_a}} a}{\sqrt {-2 \,{\mathrm e}^{2 a \textit {\_a}} \left (2 \,{\mathrm e}^{2 a \textit {\_a}} a b \textit {\_a} -4 c_{1} a^{2}-{\mathrm e}^{2 a \textit {\_a}} b \right )}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {2 \,{\mathrm e}^{2 a \textit {\_a}} a}{\sqrt {-2 \,{\mathrm e}^{2 a \textit {\_a}} \left (2 \,{\mathrm e}^{2 a \textit {\_a}} a b \textit {\_a} -4 c_{1} a^{2}-{\mathrm e}^{2 a \textit {\_a}} b \right )}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }+a {y^{\prime }}^{2}+b \sin \left (y\right ) = 0 \]

\[ \int _{}^{y}\frac {{\mathrm e}^{2 a \textit {\_a}} \left (4 a^{2}+1\right )}{\sqrt {-2 \,{\mathrm e}^{2 a \textit {\_a}} \left (4 a^{2}+1\right ) \left (2 \sin \left (\textit {\_a} \right ) {\mathrm e}^{2 a \textit {\_a}} a b -\cos \left (\textit {\_a} \right ) {\mathrm e}^{2 a \textit {\_a}} b -4 c_{1} a^{2}-c_{1} \right )}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {{\mathrm e}^{2 a \textit {\_a}} \left (4 a^{2}+1\right )}{\sqrt {-2 \,{\mathrm e}^{2 a \textit {\_a}} \left (4 a^{2}+1\right ) \left (2 \sin \left (\textit {\_a} \right ) {\mathrm e}^{2 a \textit {\_a}} a b -\cos \left (\textit {\_a} \right ) {\mathrm e}^{2 a \textit {\_a}} b -4 c_{1} a^{2}-c_{1} \right )}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }+a y {y^{\prime }}^{2}+b y = 0 \]

\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (-\left (a \,\textit {\_Z}^{2}+b \right )^{\frac {1}{2 a}}+c_{2} {\mathrm e}^{-\frac {\textit {\_a}^{2}}{2}+c_{1}}\right )}d \textit {\_a} = x +c_{3} \] Warning, solution could not be verified

\[ {}y^{\prime \prime }+a y \left (1+{y^{\prime }}^{2}\right )^{2} = 0 \]

\[ \int _{}^{y}-\frac {\textit {\_a}^{2} a -2 c_{1}}{\sqrt {-\left (\textit {\_a}^{2} a -2 c_{1} \right ) \left (\textit {\_a}^{2} a -2 c_{1} -1\right )}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}\frac {\textit {\_a}^{2} a -2 c_{1}}{\sqrt {-\left (\textit {\_a}^{2} a -2 c_{1} \right ) \left (\textit {\_a}^{2} a -2 c_{1} -1\right )}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime } = a \sqrt {1+{y^{\prime }}^{2}}+b \]

\[ -\frac {b \left (\frac {\ln \left (\frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}+b^{2}}{\sqrt {b^{2}}}+\sqrt {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )^{2} b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}-a^{2}+b^{2}}\right )}{\sqrt {b^{2}}}-\frac {\ln \left (\frac {-2 a^{2}+2 b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}+2 \sqrt {-a^{2}+b^{2}}\, \sqrt {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )^{2} b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}-a^{2}+b^{2}}}{\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )}\right )}{\sqrt {-a^{2}+b^{2}}}\right )}{a} = x +c_{2} \] Warning, solution could not be verified

\[ \frac {b \left (\frac {\ln \left (\frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}+b^{2}}{\sqrt {b^{2}}}+\sqrt {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )^{2} b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}-a^{2}+b^{2}}\right )}{\sqrt {b^{2}}}-\frac {\ln \left (\frac {-2 a^{2}+2 b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}+2 \sqrt {-a^{2}+b^{2}}\, \sqrt {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )^{2} b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}-a^{2}+b^{2}}}{\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )}\right )}{\sqrt {-a^{2}+b^{2}}}\right )}{a} = x +c_{3} \] Warning, solution could not be verified

\[ {}y^{\prime \prime } = a \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} \]

\[ -\frac {\sqrt {1-\left (y+c_{1} \right )^{2} a^{2}}}{a} = x +c_{2} \] Verified OK.

\[ \frac {\sqrt {1-\left (y+c_{1} \right )^{2} a^{2}}}{a} = x +c_{3} \] Verified OK.

\[ {}y^{\prime \prime }-a y \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} = 0 \]

\[ \int _{}^{y}\frac {\textit {\_a}^{2} a +2 c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-4 c_{1} \textit {\_a}^{2} a -4 c_{1}^{2}+4}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {\textit {\_a}^{2} a +2 c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-4 c_{1} \textit {\_a}^{2} a -4 c_{1}^{2}+4}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}8 y^{\prime \prime }+9 {y^{\prime }}^{4} = 0 \]

\[ -\left (y+c_{1} \right )^{\frac {3}{2}} = x +c_{2} \] Verified OK.

\[ \left (y+c_{1} \right )^{\frac {3}{2}} = x +c_{3} \] Verified OK.

\[ {}x y^{\prime \prime }+\left (-1+y\right ) y^{\prime } = 0 \]

\[ -\frac {2 \,\operatorname {arctanh}\left (\frac {-2 y+4}{2 \sqrt {2 c_{1} +4}}\right )}{\sqrt {2 c_{1} +4}}-\ln \left (x \right )-c_{2} = 0 \] Verified OK.

\[ {}y y^{\prime \prime }-a = 0 \]

\[ \int _{}^{y}\frac {1}{\sqrt {2 a \ln \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {2 a \ln \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}-a = 0 \]

\[ \frac {y^{2}}{2}-\frac {a \,x^{2}}{4}-\frac {c_{1} x}{2}-c_{2} = 0 \] Verified OK.

\[ {}y y^{\prime \prime }-{y^{\prime }}^{2}-1 = 0 \]

\[ \frac {\operatorname {arctanh}\left (\frac {\sqrt {-1+c_{2}^{2} {\mathrm e}^{2 c_{1}} y^{2}}}{\sqrt {{\mathrm e}^{2 c_{1}}}\, c_{2} y}\right )}{\sqrt {{\mathrm e}^{2 c_{1}}}\, c_{2}} = x +c_{3} \] Verified OK.

\[ -\frac {\operatorname {arctanh}\left (\frac {\sqrt {-1+c_{2}^{2} {\mathrm e}^{2 c_{1}} y^{2}}}{\sqrt {{\mathrm e}^{2 c_{1}}}\, c_{2} y}\right )}{\sqrt {{\mathrm e}^{2 c_{1}}}\, c_{2}} = x +c_{4} \] Verified OK.

\[ {}y y^{\prime \prime }-{y^{\prime }}^{2}+a y y^{\prime }+b y^{2} = 0 \]

\[ \int _{}^{y}\frac {1}{\frac {b \textit {\_a} \left (-\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {\ln \left (\textit {\_a} \right ) a^{2}-c_{2} a^{2}-b}{b}}}{b}\right )+\frac {\ln \left (\textit {\_a} \right ) a^{2}-c_{2} a^{2}-b}{b}\right )}{a}-\textit {\_a} a \left (\ln \left (\textit {\_a} \right )-c_{2} \right )}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y y^{\prime \prime }-3 {y^{\prime }}^{2}+3 y y^{\prime }-y^{2} = 0 \]

\[ \int _{}^{y}\frac {1}{\frac {\textit {\_a}^{3} c_{3}^{2}}{2}+\frac {\textit {\_a}^{2} c_{3} \sqrt {2}\, \sqrt {2 c_{3}^{2} \textit {\_a}^{2}+4}}{4}+\textit {\_a}}d \textit {\_a} = x +c_{4} \] Verified OK.

\[ \int _{}^{y}\frac {1}{\frac {\textit {\_a}^{3} c_{3}^{2}}{2}-\frac {\textit {\_a}^{2} c_{3} \sqrt {2}\, \sqrt {2 c_{3}^{2} \textit {\_a}^{2}+4}}{4}+\textit {\_a}}d \textit {\_a} = x +c_{5} \] Verified OK.

\[ {}y y^{\prime \prime }+a \left (1+{y^{\prime }}^{2}\right ) = 0 \]

\[ \int _{}^{y}\frac {\textit {\_a}^{a}}{\sqrt {c_{2}^{2} {\mathrm e}^{2 c_{1}}-\textit {\_a}^{2 a}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ \int _{}^{y}-\frac {\textit {\_a}^{a}}{\sqrt {c_{2}^{2} {\mathrm e}^{2 c_{1}}-\textit {\_a}^{2 a}}}d \textit {\_a} = x +c_{4} \] Verified OK.

\[ {}y y^{\prime \prime }+a {y^{\prime }}^{2}+b y^{3} = 0 \]

\[ \int _{}^{y}\frac {\left (3+2 a \right ) \textit {\_a}^{2 a}}{\sqrt {-2 \left (3+2 a \right ) \textit {\_a}^{2 a} \left (\textit {\_a}^{3+2 a} b -2 c_{1} a -3 c_{1} \right )}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {\left (3+2 a \right ) \textit {\_a}^{2 a}}{\sqrt {-2 \left (3+2 a \right ) \textit {\_a}^{2 a} \left (\textit {\_a}^{3+2 a} b -2 c_{1} a -3 c_{1} \right )}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}y y^{\prime \prime }+a {y^{\prime }}^{2}+b y^{2} y^{\prime }+c y^{4} = 0 \]

\[ \int _{}^{y}\frac {4+2 a}{\textit {\_a}^{2} \left (\tan \left (\operatorname {RootOf}\left (-2 a \ln \left (\textit {\_a} \right ) \sqrt {4 a c -b^{2}+8 c}+2 c_{1} \sqrt {4 a c -b^{2}+8 c}\, a -\sqrt {4 a c -b^{2}+8 c}\, \ln \left (\frac {\textit {\_a}^{4} \left (4 a c \tan \left (\textit {\_Z} \right )^{2}-b^{2} \tan \left (\textit {\_Z} \right )^{2}+8 c \tan \left (\textit {\_Z} \right )^{2}+4 a c -b^{2}+8 c \right )}{4 a +8}\right )+4 c_{1} \sqrt {4 a c -b^{2}+8 c}+2 \textit {\_Z} b \right )\right ) \sqrt {4 a c -b^{2}+8 c}-b \right )}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ {}y y^{\prime \prime }-{y^{\prime }}^{2}-1-2 a y \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} = 0 \]

\[ \int _{}^{y}\frac {\textit {\_a}^{2} a +c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-2 c_{1} \textit {\_a}^{2} a -c_{1}^{2}+\textit {\_a}^{2}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ \int _{}^{y}-\frac {\textit {\_a}^{2} a +c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-2 c_{1} \textit {\_a}^{2} a -c_{1}^{2}+\textit {\_a}^{2}}}d \textit {\_a} = x +c_{4} \] Verified OK.

\[ {}y^{\prime \prime } \left (x +y\right )+{y^{\prime }}^{2}-y^{\prime } = 0 \]

\[ -\ln \left (2 y+c_{1} \right )+2 \ln \left (c_{1} -x +y\right ) = c_{1} \] Verified OK.

\[ {}2 y y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \]

\[ -\sqrt {y \,{\mathrm e}^{2 c_{1}} c_{2}^{2}-y^{2}}+\frac {c_{2}^{2} {\mathrm e}^{2 c_{1}} \arctan \left (\frac {y-\frac {c_{2}^{2} {\mathrm e}^{2 c_{1}}}{2}}{\sqrt {y \,{\mathrm e}^{2 c_{1}} c_{2}^{2}-y^{2}}}\right )}{2} = x +c_{3} \] Verified OK.

\[ \sqrt {y \,{\mathrm e}^{2 c_{1}} c_{2}^{2}-y^{2}}-\frac {c_{2}^{2} {\mathrm e}^{2 c_{1}} \arctan \left (\frac {y-\frac {c_{2}^{2} {\mathrm e}^{2 c_{1}}}{2}}{\sqrt {y \,{\mathrm e}^{2 c_{1}} c_{2}^{2}-y^{2}}}\right )}{2} = x +c_{4} \] Verified OK.

\[ {}2 y y^{\prime \prime }-{y^{\prime }}^{2}-8 y^{3} = 0 \]

\[ \int _{}^{y}\frac {1}{\sqrt {4 \textit {\_a}^{3}+2 c_{1} \textit {\_a}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {4 \textit {\_a}^{3}+2 c_{1} \textit {\_a}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}2 y y^{\prime \prime }-{y^{\prime }}^{2}-8 y^{3}-4 y^{2} = 0 \]

\[ \int _{}^{y}\frac {1}{\sqrt {4 \textit {\_a}^{3}+2 c_{1} \textit {\_a} +4 \textit {\_a}^{2}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {4 \textit {\_a}^{3}+2 c_{1} \textit {\_a} +4 \textit {\_a}^{2}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}2 y y^{\prime \prime }-{y^{\prime }}^{2}+\left (a y+b \right ) y^{2} = 0 \]

\[ \int _{}^{y}\frac {2}{\sqrt {-2 \textit {\_a}^{3} a -4 \textit {\_a}^{2} b +8 c_{1} \textit {\_a}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {2}{\sqrt {-2 \textit {\_a}^{3} a -4 \textit {\_a}^{2} b +8 c_{1} \textit {\_a}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}2 y y^{\prime \prime }-{y^{\prime }}^{2}-3 y^{4} = 0 \]

\[ \int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{4}+2 c_{1} \textit {\_a}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {\textit {\_a}^{4}+2 c_{1} \textit {\_a}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}2 y y^{\prime \prime }-6 {y^{\prime }}^{2}+\left (1+a y^{3}\right ) y^{2} = 0 \]

\[ \int _{}^{y}\frac {2}{\sqrt {8 c_{1} \textit {\_a}^{4}+4 \textit {\_a}^{3} a +1}\, \textit {\_a}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {2}{\sqrt {8 c_{1} \textit {\_a}^{4}+4 \textit {\_a}^{3} a +1}\, \textit {\_a}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}2 y y^{\prime \prime }-{y^{\prime }}^{2} \left (1+{y^{\prime }}^{2}\right ) = 0 \]

\[ \int _{}^{y}\frac {1}{c_{2} \sqrt {\textit {\_a}}\, \sqrt {-\frac {1}{c_{2}^{2} \textit {\_a} -1}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}2 \left (y-a \right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \]

\[ \frac {\left (-y+a \right ) \left (c_{2}^{2} {\mathrm e}^{2 c_{1}}+y-a \right )}{\sqrt {-\left (y-a \right ) \left (c_{2}^{2} {\mathrm e}^{2 c_{1}}+y-a \right )}}+\frac {c_{2}^{2} {\mathrm e}^{2 c_{1}} \arctan \left (\frac {y+\frac {c_{2}^{2} {\mathrm e}^{2 c_{1}}}{2}-a}{\sqrt {-y^{2}+\left (-c_{2}^{2} {\mathrm e}^{2 c_{1}}+2 a \right ) y+a \left (c_{2}^{2} {\mathrm e}^{2 c_{1}}-a \right )}}\right )}{2} = x +c_{3} \] Verified OK.

\[ -\frac {\left (-y+a \right ) \left (c_{2}^{2} {\mathrm e}^{2 c_{1}}+y-a \right )}{\sqrt {-\left (y-a \right ) \left (c_{2}^{2} {\mathrm e}^{2 c_{1}}+y-a \right )}}-\frac {c_{2}^{2} {\mathrm e}^{2 c_{1}} \arctan \left (\frac {y+\frac {c_{2}^{2} {\mathrm e}^{2 c_{1}}}{2}-a}{\sqrt {-y^{2}+\left (-c_{2}^{2} {\mathrm e}^{2 c_{1}}+2 a \right ) y+a \left (c_{2}^{2} {\mathrm e}^{2 c_{1}}-a \right )}}\right )}{2} = x +c_{4} \] Verified OK.

\[ {}4 y y^{\prime \prime }-3 {y^{\prime }}^{2}-12 y^{3} = 0 \]

\[ \int _{}^{y}\frac {1}{\sqrt {2 c_{1} \textit {\_a}^{\frac {3}{2}}+4 \textit {\_a}^{3}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {1}{\sqrt {2 c_{1} \textit {\_a}^{\frac {3}{2}}+4 \textit {\_a}^{3}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}4 y y^{\prime \prime }-3 {y^{\prime }}^{2}+a y^{3}+b y^{2}+c y = 0 \]

\[ \int _{}^{y}\frac {3}{\sqrt {-3 a \,\textit {\_a}^{3}+18 c_{1} \textit {\_a}^{\frac {3}{2}}-9 \textit {\_a}^{2} b +9 c \textit {\_a}}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {3}{\sqrt {-3 a \,\textit {\_a}^{3}+18 c_{1} \textit {\_a}^{\frac {3}{2}}-9 \textit {\_a}^{2} b +9 c \textit {\_a}}}d \textit {\_a} = x +c_{3} \] Verified OK.

\[ {}4 y y^{\prime \prime }-5 {y^{\prime }}^{2}+a y^{2} = 0 \]

\[ -\frac {4 y \sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}}{\sqrt {a}}\right )}{\sqrt {y^{\frac {5}{2}} {\mathrm e}^{2 c_{2}} c_{3}^{2}+y^{2} a}\, \sqrt {a}} = x +c_{4} \] Verified OK.

\[ \frac {4 y \sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}}{\sqrt {a}}\right )}{\sqrt {y^{\frac {5}{2}} {\mathrm e}^{2 c_{2}} c_{3}^{2}+y^{2} a}\, \sqrt {a}} = x +c_{5} \] Verified OK.

\[ {}a y y^{\prime \prime }+b {y^{\prime }}^{2}+\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y+\operatorname {c0} = 0 \]

\[ \int _{}^{y}\frac {\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 a^{2} b^{2}+20 a \,b^{3}+4 b^{4}\right )}{\sqrt {-\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 a^{2} b^{2}+20 a \,b^{3}+4 b^{4}\right ) \left (-12 c_{1} a^{4} b -50 c_{1} a^{3} b^{2}-70 c_{1} a^{2} b^{3}-40 c_{1} a \,b^{4}+12 b \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{3}+26 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{2} b^{2}+18 a \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{3}+16 a \operatorname {c2} \,b^{3} \textit {\_a}^{\frac {2 b +2 a}{a}}+3 b \operatorname {c4} \,a^{3} \textit {\_a}^{\frac {2 b +4 a}{a}}+11 \operatorname {c4} \,a^{2} b^{2} \textit {\_a}^{\frac {2 b +4 a}{a}}+12 a \operatorname {c4} \,b^{3} \textit {\_a}^{\frac {2 b +4 a}{a}}+6 b \operatorname {c2} \,a^{3} \textit {\_a}^{\frac {2 b +2 a}{a}}+19 \operatorname {c2} \,a^{2} b^{2} \textit {\_a}^{\frac {2 b +2 a}{a}}+25 \textit {\_a}^{\frac {2 b}{a}} a^{3} b \operatorname {c0} +35 \textit {\_a}^{\frac {2 b}{a}} a^{2} b^{2} \operatorname {c0} +20 \textit {\_a}^{\frac {2 b}{a}} a \,b^{3} \operatorname {c0} +4 b \operatorname {c3} \,a^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+14 \operatorname {c3} \,a^{2} b^{2} \textit {\_a}^{\frac {3 a +2 b}{a}}+14 a \operatorname {c3} \,b^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+4 \textit {\_a}^{\frac {2 b}{a}} b^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{4}+4 \operatorname {c3} \,b^{4} \textit {\_a}^{\frac {3 a +2 b}{a}}+4 \operatorname {c2} \,b^{4} \textit {\_a}^{\frac {2 b +2 a}{a}}+4 \operatorname {c4} \,b^{4} \textit {\_a}^{\frac {2 b +4 a}{a}}-8 c_{1} b^{5}+6 \textit {\_a}^{\frac {2 b}{a}} a^{4} \operatorname {c0} \right )}}d \textit {\_a} = x +c_{2} \] Verified OK.

\[ \int _{}^{y}-\frac {\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 a^{2} b^{2}+20 a \,b^{3}+4 b^{4}\right )}{\sqrt {-\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 a^{2} b^{2}+20 a \,b^{3}+4 b^{4}\right ) \left (-12 c_{1} a^{4} b -50 c_{1} a^{3} b^{2}-70 c_{1} a^{2} b^{3}-40 c_{1} a \,b^{4}+6 \textit {\_a}^{\frac {2 b}{a}} a^{4} \operatorname {c0} +25 \textit {\_a}^{\frac {2 b}{a}} a^{3} b \operatorname {c0} +35 \textit {\_a}^{\frac {2 b}{a}} a^{2} b^{2} \operatorname {c0} +20 \textit {\_a}^{\frac {2 b}{a}} a \,b^{3} \operatorname {c0} +4 b \operatorname {c3} \,a^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+14 \operatorname {c3} \,a^{2} b^{2} \textit {\_a}^{\frac {3 a +2 b}{a}}+14 a \operatorname {c3} \,b^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+12 b \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{3}+26 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{2} b^{2}+18 a \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{3}+3 b \operatorname {c4} \,a^{3} \textit {\_a}^{\frac {2 b +4 a}{a}}+11 \operatorname {c4} \,a^{2} b^{2} \textit {\_a}^{\frac {2 b +4 a}{a}}+12 a \operatorname {c4} \,b^{3} \textit {\_a}^{\frac {2 b +4 a}{a}}+6 b \operatorname {c2} \,a^{3} \textit {\_a}^{\frac {2 b +2 a}{a}}+19 \operatorname {c2} \,a^{2} b^{2} \textit {\_a}^{\frac {2 b +2 a}{a}}+16 a \operatorname {c2} \,b^{3} \textit {\_a}^{\frac {2 b +2 a}{a}}-8 c_{1} b^{5}+4 \textit {\_a}^{\frac {2 b}{a}} b^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{4}+4 \operatorname {c3} \,b^{4} \textit {\_a}^{\frac {3 a +2 b}{a}}+4 \operatorname {c2} \,b^{4} \textit {\_a}^{\frac {2 b +2 a}{a}}+4 \operatorname {c4} \,b^{4} \textit {\_a}^{\frac {2 b +4 a}{a}}\right )}}d \textit {\_a} = x +c_{3} \] Verified OK.