#### 2.489   ODE No. 489

$a y(x)^2+b x+c+y(x)^2 y'(x)^2+2 x y(x) y'(x)=0$ Mathematica : cpu = 300.213 (sec), leaf count = 0 , timed out

\$Aborted

Maple : cpu = 3.863 (sec), leaf count = 1101

$\left \{ y \left ( x \right ) =-{\frac {\sqrt {16}}{2\,a \left ( a+1 \right ) }\sqrt {a \left ( a \left ( a+1 \right ) ^{2} \left ( ax-{\frac {b}{2}}+x \right ) ^{2}{\it RootOf} \left ( -b\ln \left ( 2\,ax-b+2\,x \right ) +2\,\int ^{{\it \_Z}}\!1/4\,{\frac {b}{ \left ( a+1 \right ) \left ( 4\,{\it \_a}\,{a}^{2}+8\,{\it \_a}\,a+4\,{\it \_a}+a+2 \right ) {\it \_a}} \left ( -4\,{\it \_a}\,{a}^{2}+\sqrt {- \left ( 4\,{\it \_a}\,{a}^{3}+8\,{\it \_a}\,{a}^{2}+4\,{\it \_a}\,a-1 \right ) {{\rm e}^{4\,{\frac {a+1}{b}}}}}{{\rm e}^{-2\,{\frac {a+1}{b}}}}-8\,{\it \_a}\,a-4\,{\it \_a}-1 \right ) }{d{\it \_a}}a+2\,{\it \_C1}\,a+2\,\int ^{{\it \_Z}}\!1/4\,{\frac {b}{ \left ( a+1 \right ) \left ( 4\,{\it \_a}\,{a}^{2}+8\,{\it \_a}\,a+4\,{\it \_a}+a+2 \right ) {\it \_a}} \left ( -4\,{\it \_a}\,{a}^{2}+\sqrt {- \left ( 4\,{\it \_a}\,{a}^{3}+8\,{\it \_a}\,{a}^{2}+4\,{\it \_a}\,a-1 \right ) {{\rm e}^{4\,{\frac {a+1}{b}}}}}{{\rm e}^{-2\,{\frac {a+1}{b}}}}-8\,{\it \_a}\,a-4\,{\it \_a}-1 \right ) }{d{\it \_a}}+2\,{\it \_C1} \right ) + \left ( -{\frac {bx}{4}}-{\frac {c}{4}} \right ) {a}^{2}+ \left ( -{\frac {bx}{4}}-{\frac {c}{2}} \right ) a-{\frac {{b}^{2}}{16}}-{\frac {c}{4}} \right ) }},y \left ( x \right ) ={\frac {\sqrt {16}}{2\,a \left ( a+1 \right ) }\sqrt {a \left ( a \left ( a+1 \right ) ^{2} \left ( ax-{\frac {b}{2}}+x \right ) ^{2}{\it RootOf} \left ( -b\ln \left ( 2\,ax-b+2\,x \right ) +2\,\int ^{{\it \_Z}}\!1/4\,{\frac {b}{ \left ( a+1 \right ) \left ( 4\,{\it \_a}\,{a}^{2}+8\,{\it \_a}\,a+4\,{\it \_a}+a+2 \right ) {\it \_a}} \left ( -4\,{\it \_a}\,{a}^{2}+\sqrt {- \left ( 4\,{\it \_a}\,{a}^{3}+8\,{\it \_a}\,{a}^{2}+4\,{\it \_a}\,a-1 \right ) {{\rm e}^{4\,{\frac {a+1}{b}}}}}{{\rm e}^{-2\,{\frac {a+1}{b}}}}-8\,{\it \_a}\,a-4\,{\it \_a}-1 \right ) }{d{\it \_a}}a+2\,{\it \_C1}\,a+2\,\int ^{{\it \_Z}}\!1/4\,{\frac {b}{ \left ( a+1 \right ) \left ( 4\,{\it \_a}\,{a}^{2}+8\,{\it \_a}\,a+4\,{\it \_a}+a+2 \right ) {\it \_a}} \left ( -4\,{\it \_a}\,{a}^{2}+\sqrt {- \left ( 4\,{\it \_a}\,{a}^{3}+8\,{\it \_a}\,{a}^{2}+4\,{\it \_a}\,a-1 \right ) {{\rm e}^{4\,{\frac {a+1}{b}}}}}{{\rm e}^{-2\,{\frac {a+1}{b}}}}-8\,{\it \_a}\,a-4\,{\it \_a}-1 \right ) }{d{\it \_a}}+2\,{\it \_C1} \right ) + \left ( -{\frac {bx}{4}}-{\frac {c}{4}} \right ) {a}^{2}+ \left ( -{\frac {bx}{4}}-{\frac {c}{2}} \right ) a-{\frac {{b}^{2}}{16}}-{\frac {c}{4}} \right ) }},y \left ( x \right ) =-{\frac {\sqrt {16}}{2\,a \left ( a+1 \right ) }\sqrt {a \left ( a \left ( a+1 \right ) ^{2} \left ( ax-{\frac {b}{2}}+x \right ) ^{2}{\it RootOf} \left ( -b\ln \left ( 2\,ax-b+2\,x \right ) +2\,\int ^{{\it \_Z}}\!-1/4\,{\frac {b}{ \left ( a+1 \right ) \left ( 4\,{\it \_a}\,{a}^{2}+8\,{\it \_a}\,a+4\,{\it \_a}+a+2 \right ) {\it \_a}} \left ( 4\,{\it \_a}\,{a}^{2}+\sqrt {- \left ( 4\,{\it \_a}\,{a}^{3}+8\,{\it \_a}\,{a}^{2}+4\,{\it \_a}\,a-1 \right ) {{\rm e}^{4\,{\frac {a+1}{b}}}}}{{\rm e}^{-2\,{\frac {a+1}{b}}}}+8\,{\it \_a}\,a+4\,{\it \_a}+1 \right ) }{d{\it \_a}}a+2\,{\it \_C1}\,a+2\,\int ^{{\it \_Z}}\!-1/4\,{\frac {b}{ \left ( a+1 \right ) \left ( 4\,{\it \_a}\,{a}^{2}+8\,{\it \_a}\,a+4\,{\it \_a}+a+2 \right ) {\it \_a}} \left ( 4\,{\it \_a}\,{a}^{2}+\sqrt {- \left ( 4\,{\it \_a}\,{a}^{3}+8\,{\it \_a}\,{a}^{2}+4\,{\it \_a}\,a-1 \right ) {{\rm e}^{4\,{\frac {a+1}{b}}}}}{{\rm e}^{-2\,{\frac {a+1}{b}}}}+8\,{\it \_a}\,a+4\,{\it \_a}+1 \right ) }{d{\it \_a}}+2\,{\it \_C1} \right ) + \left ( -{\frac {bx}{4}}-{\frac {c}{4}} \right ) {a}^{2}+ \left ( -{\frac {bx}{4}}-{\frac {c}{2}} \right ) a-{\frac {{b}^{2}}{16}}-{\frac {c}{4}} \right ) }},y \left ( x \right ) ={\frac {\sqrt {16}}{2\,a \left ( a+1 \right ) }\sqrt {a \left ( a \left ( a+1 \right ) ^{2} \left ( ax-{\frac {b}{2}}+x \right ) ^{2}{\it RootOf} \left ( -b\ln \left ( 2\,ax-b+2\,x \right ) +2\,\int ^{{\it \_Z}}\!-1/4\,{\frac {b}{ \left ( a+1 \right ) \left ( 4\,{\it \_a}\,{a}^{2}+8\,{\it \_a}\,a+4\,{\it \_a}+a+2 \right ) {\it \_a}} \left ( 4\,{\it \_a}\,{a}^{2}+\sqrt {- \left ( 4\,{\it \_a}\,{a}^{3}+8\,{\it \_a}\,{a}^{2}+4\,{\it \_a}\,a-1 \right ) {{\rm e}^{4\,{\frac {a+1}{b}}}}}{{\rm e}^{-2\,{\frac {a+1}{b}}}}+8\,{\it \_a}\,a+4\,{\it \_a}+1 \right ) }{d{\it \_a}}a+2\,{\it \_C1}\,a+2\,\int ^{{\it \_Z}}\!-1/4\,{\frac {b}{ \left ( a+1 \right ) \left ( 4\,{\it \_a}\,{a}^{2}+8\,{\it \_a}\,a+4\,{\it \_a}+a+2 \right ) {\it \_a}} \left ( 4\,{\it \_a}\,{a}^{2}+\sqrt {- \left ( 4\,{\it \_a}\,{a}^{3}+8\,{\it \_a}\,{a}^{2}+4\,{\it \_a}\,a-1 \right ) {{\rm e}^{4\,{\frac {a+1}{b}}}}}{{\rm e}^{-2\,{\frac {a+1}{b}}}}+8\,{\it \_a}\,a+4\,{\it \_a}+1 \right ) }{d{\it \_a}}+2\,{\it \_C1} \right ) + \left ( -{\frac {bx}{4}}-{\frac {c}{4}} \right ) {a}^{2}+ \left ( -{\frac {bx}{4}}-{\frac {c}{2}} \right ) a-{\frac {{b}^{2}}{16}}-{\frac {c}{4}} \right ) }} \right \}$