\[ y''(x)-\frac {1}{\left (a y(x)^2+b x y(x)+c x^2+d y(x)+e x+k\right )^{3/2}}=0 \] ✗ Mathematica : cpu = 60.3594 (sec), leaf count = 0 , could not solve
DSolve[-(k + e*x + c*x^2 + d*y[x] + b*x*y[x] + a*y[x]^2)^(-3/2) + Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 31.087 (sec), leaf count = 7543
\[ \left \{ y \left ( x \right ) ={\frac {1}{2\,a} \left ( 2\,{\it RootOf} \left ( -4\,\arctan \left ( 1/2\,{\frac {4\,acx-{b}^{2}x+2\,a\beta -b\alpha }{\sqrt {a \left ( -a{\beta }^{2}+4\,ac\gamma -{\alpha }^{2}c+\alpha \,b\beta -{b}^{2}\gamma \right ) }}} \right ) ac+\arctan \left ( 1/2\,{\frac {4\,acx-{b}^{2}x+2\,a\beta -b\alpha }{\sqrt {a \left ( -a{\beta }^{2}+4\,ac\gamma -{\alpha }^{2}c+\alpha \,b\beta -{b}^{2}\gamma \right ) }}} \right ) {b}^{2}-2\,\int ^{{\it \_Z}}\!1/4\,{\sqrt {16} \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}\sqrt {-{\frac {\sqrt {16}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}} \left ( 8\,{\alpha }^{3}ac{{\it \_g}}^{2}b\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{a}^{3}{\beta }^{3}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-64\,{\it \_C1}\,{a}^{3}{c}^{3}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+{\it \_C1}\,{\alpha }^{2}{b}^{4}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+2\,{\it \_C1}\,\alpha \,{b}^{5}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+16\,\beta \,b\alpha \,{a}^{2}{{\it \_g}}^{2}\gamma \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-32\,{a}^{2}c\gamma \,{{\it \_g}}^{2}b\alpha \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+16\,\beta \,b\alpha \,{a}^{2}{{\it \_g}}^{2}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+128\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,a\alpha \,b+64\,{a}^{3}c{\gamma }^{2}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{a}^{2}{b}^{2}{\gamma }^{2}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{a}^{3}{\beta }^{2}{{\it \_g}}^{2}\gamma \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+64\,{a}^{3}{c}^{2}\gamma \,{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,a{b}^{4}\gamma \,{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{a}^{3}{\beta }^{2}{{\it \_g}}^{2}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,{a}^{2}{\beta }^{2}{{\it \_g}}^{2}{\alpha }^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{\alpha }^{2}{a}^{2}{c}^{2}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,{a}^{2}{\beta }^{2}{{\it \_g}}^{2}{b}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,{\alpha }^{4}ac{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-64\,\gamma \,{\it \_C1}\,{a}^{3}{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,\gamma \,{\it \_C1}\,a{b}^{4}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-64\,{\it \_C1}\,{a}^{3}\beta \,{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+16\,{\it \_C1}\,{a}^{2}{\alpha }^{2}{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+48\,{\it \_C1}\,{a}^{2}{b}^{2}{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,{\it \_C1}\,a{b}^{4}\beta \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-12\,{\it \_C1}\,a{b}^{4}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-8\,{\it \_C1}\,a{\alpha }^{2}{b}^{2}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{\it \_C1}\,a\alpha \,{b}^{3}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-8\,\beta \,{b}^{2}{\alpha }^{2}a{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,{\alpha }^{2}ac{{\it \_g}}^{2}{b}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,\beta \,{b}^{3}\alpha \,a{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+32\,\gamma \,{\it \_C1}\,{a}^{2}{b}^{2}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+64\,{a}^{3}c\gamma \,{{\it \_g}}^{2}\beta \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-32\,{a}^{2}c\gamma \,{{\it \_g}}^{2}{\alpha }^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{a}^{2}{b}^{2}\gamma \,{{\it \_g}}^{2}\beta \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-32\,{a}^{2}c\gamma \,{{\it \_g}}^{2}{b}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,a{b}^{2}\gamma \,{{\it \_g}}^{2}{\alpha }^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+8\,a{b}^{3}\gamma \,{{\it \_g}}^{2}\alpha \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{\alpha }^{2}{a}^{2}c{{\it \_g}}^{2}\beta \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+24\,{a}^{2}{\beta }^{2}{{\it \_g}}^{2}b\alpha \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,\beta \,b{\alpha }^{3}a{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+32\,{\it \_C1}\,{a}^{2}\alpha \,b{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+32\,{\it \_C1}\,{a}^{2}{b}^{2}\beta \,c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-256\,\gamma \,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,{a}^{2}-256\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,{a}^{2}\beta -256\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,{a}^{2}c+64\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,a{\alpha }^{2}+64\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,a{b}^{2}+{\it \_C1}\,{b}^{6}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}} \right ) {\frac {1}{\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}}}}} \left ( 16\,{a}^{2}c\gamma \,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,a{b}^{2}\gamma \,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,{a}^{2}{\beta }^{2}\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,{\alpha }^{2}ac\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,\beta \,b\alpha \,a\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_C1}\,{a}^{2}{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+8\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_C1}\,a{b}^{2}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_C1}\,{b}^{4}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-256\,\gamma \,{\it \_g}\,{a}^{2}-256\,{\it \_g}\,{a}^{2}\beta -256\,{\it \_g}\,{a}^{2}c+64\,{\it \_g}\,a{\alpha }^{2}+128\,{\it \_g}\,a\alpha \,b+64\,{\it \_g}\,a{b}^{2} \right ) ^{-1}}{d{\it \_g}}\sqrt {a \left ( -a{\beta }^{2}+4\,ac\gamma -{\alpha }^{2}c+\alpha \,b\beta -{b}^{2}\gamma \right ) }+2\,{\it \_C2}\,\sqrt {a \left ( -a{\beta }^{2}+4\,ac\gamma -{\alpha }^{2}c+\alpha \,b\beta -{b}^{2}\gamma \right ) } \right ) a\sqrt {4\,ac{x}^{2}-{b}^{2}{x}^{2}+4\,ax\beta -2\,\alpha \,bx+4\,a\gamma -{\alpha }^{2}}-bx-\alpha \right ) },y \left ( x \right ) ={\frac {1}{2\,a} \left ( 2\,{\it RootOf} \left ( -4\,\arctan \left ( 1/2\,{\frac {4\,acx-{b}^{2}x+2\,a\beta -b\alpha }{\sqrt {a \left ( -a{\beta }^{2}+4\,ac\gamma -{\alpha }^{2}c+\alpha \,b\beta -{b}^{2}\gamma \right ) }}} \right ) ac+\arctan \left ( 1/2\,{\frac {4\,acx-{b}^{2}x+2\,a\beta -b\alpha }{\sqrt {a \left ( -a{\beta }^{2}+4\,ac\gamma -{\alpha }^{2}c+\alpha \,b\beta -{b}^{2}\gamma \right ) }}} \right ) {b}^{2}+2\,\int ^{{\it \_Z}}\!1/4\,{\sqrt {16} \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}\sqrt {-{\frac {\sqrt {16}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}} \left ( 8\,{\alpha }^{3}ac{{\it \_g}}^{2}b\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{a}^{3}{\beta }^{3}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-64\,{\it \_C1}\,{a}^{3}{c}^{3}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+{\it \_C1}\,{\alpha }^{2}{b}^{4}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+2\,{\it \_C1}\,\alpha \,{b}^{5}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+16\,\beta \,b\alpha \,{a}^{2}{{\it \_g}}^{2}\gamma \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-32\,{a}^{2}c\gamma \,{{\it \_g}}^{2}b\alpha \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+16\,\beta \,b\alpha \,{a}^{2}{{\it \_g}}^{2}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+128\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,a\alpha \,b+64\,{a}^{3}c{\gamma }^{2}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{a}^{2}{b}^{2}{\gamma }^{2}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{a}^{3}{\beta }^{2}{{\it \_g}}^{2}\gamma \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+64\,{a}^{3}{c}^{2}\gamma \,{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,a{b}^{4}\gamma \,{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{a}^{3}{\beta }^{2}{{\it \_g}}^{2}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,{a}^{2}{\beta }^{2}{{\it \_g}}^{2}{\alpha }^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{\alpha }^{2}{a}^{2}{c}^{2}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,{a}^{2}{\beta }^{2}{{\it \_g}}^{2}{b}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,{\alpha }^{4}ac{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-64\,\gamma \,{\it \_C1}\,{a}^{3}{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,\gamma \,{\it \_C1}\,a{b}^{4}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-64\,{\it \_C1}\,{a}^{3}\beta \,{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+16\,{\it \_C1}\,{a}^{2}{\alpha }^{2}{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+48\,{\it \_C1}\,{a}^{2}{b}^{2}{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,{\it \_C1}\,a{b}^{4}\beta \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-12\,{\it \_C1}\,a{b}^{4}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-8\,{\it \_C1}\,a{\alpha }^{2}{b}^{2}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{\it \_C1}\,a\alpha \,{b}^{3}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-8\,\beta \,{b}^{2}{\alpha }^{2}a{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,{\alpha }^{2}ac{{\it \_g}}^{2}{b}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,\beta \,{b}^{3}\alpha \,a{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+32\,\gamma \,{\it \_C1}\,{a}^{2}{b}^{2}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+64\,{a}^{3}c\gamma \,{{\it \_g}}^{2}\beta \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-32\,{a}^{2}c\gamma \,{{\it \_g}}^{2}{\alpha }^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{a}^{2}{b}^{2}\gamma \,{{\it \_g}}^{2}\beta \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-32\,{a}^{2}c\gamma \,{{\it \_g}}^{2}{b}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,a{b}^{2}\gamma \,{{\it \_g}}^{2}{\alpha }^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+8\,a{b}^{3}\gamma \,{{\it \_g}}^{2}\alpha \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,{\alpha }^{2}{a}^{2}c{{\it \_g}}^{2}\beta \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+24\,{a}^{2}{\beta }^{2}{{\it \_g}}^{2}b\alpha \,\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,\beta \,b{\alpha }^{3}a{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+32\,{\it \_C1}\,{a}^{2}\alpha \,b{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+32\,{\it \_C1}\,{a}^{2}{b}^{2}\beta \,c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-256\,\gamma \,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,{a}^{2}-256\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,{a}^{2}\beta -256\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,{a}^{2}c+64\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,a{\alpha }^{2}+64\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_g}\,a{b}^{2}+{\it \_C1}\,{b}^{6}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}} \right ) {\frac {1}{\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}}}}} \left ( 16\,{a}^{2}c\gamma \,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,a{b}^{2}\gamma \,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,{a}^{2}{\beta }^{2}\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-4\,{\alpha }^{2}ac\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+4\,\beta \,b\alpha \,a\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{{\it \_g}}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-16\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_C1}\,{a}^{2}{c}^{2}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}+8\,\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_C1}\,a{b}^{2}c\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-\sqrt {4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2}}{\it \_C1}\,{b}^{4}\sqrt {16}\sqrt {{\frac { \left ( 4\,{{\it \_g}}^{2}{a}^{2}+1 \right ) \left ( 4\,a\beta +4\,ac+4\,a\gamma -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{a}}}-256\,\gamma \,{\it \_g}\,{a}^{2}-256\,{\it \_g}\,{a}^{2}\beta -256\,{\it \_g}\,{a}^{2}c+64\,{\it \_g}\,a{\alpha }^{2}+128\,{\it \_g}\,a\alpha \,b+64\,{\it \_g}\,a{b}^{2} \right ) ^{-1}}{d{\it \_g}}\sqrt {a \left ( -a{\beta }^{2}+4\,ac\gamma -{\alpha }^{2}c+\alpha \,b\beta -{b}^{2}\gamma \right ) }+2\,{\it \_C2}\,\sqrt {a \left ( -a{\beta }^{2}+4\,ac\gamma -{\alpha }^{2}c+\alpha \,b\beta -{b}^{2}\gamma \right ) } \right ) a\sqrt {4\,ac{x}^{2}-{b}^{2}{x}^{2}+4\,ax\beta -2\,\alpha \,bx+4\,a\gamma -{\alpha }^{2}}-bx-\alpha \right ) } \right \} \]