\[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx \]
Optimal antiderivative \[ \frac {\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}, \sqrt {\frac {-b -\sqrt {-4 a c +b^{2}}}{b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {b +\sqrt {-4 a c +b^{2}}}\, \sqrt {2}}{2 \sqrt {c}} \]
command
int((1+2*c*x^2/(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(1-2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))^(1/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| elliptic | \(\frac {\sqrt {\frac {-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b}{-b +\sqrt {-4 a c +b^{2}}}}\, \left (-b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {-\frac {\left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right ) \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{a c}}\, \left (\frac {\sqrt {2}\, \sqrt {1-\frac {2 c \,x^{2}}{-b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {1-\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (x \sqrt {2}\, \sqrt {\frac {c}{-b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-4+\frac {2 \left (-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c}{b -\sqrt {-4 a c +b^{2}}}\right ) \left (b -\sqrt {-4 a c +b^{2}}\right )}{c}}}{2}\right )}{2 \sqrt {\frac {c}{-b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {1-\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}-\frac {4 c^{2} x^{4}}{\left (b -\sqrt {-4 a c +b^{2}}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {2 c \sqrt {2}\, \sqrt {1-\frac {2 c \,x^{2}}{-b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {1-\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\EllipticF \left (x \sqrt {2}\, \sqrt {\frac {c}{-b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-4+\frac {2 \left (-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c}{b -\sqrt {-4 a c +b^{2}}}\right ) \left (b -\sqrt {-4 a c +b^{2}}\right )}{c}}}{2}\right )-\EllipticE \left (x \sqrt {2}\, \sqrt {\frac {c}{-b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-4+\frac {2 \left (-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c}{b -\sqrt {-4 a c +b^{2}}}\right ) \left (b -\sqrt {-4 a c +b^{2}}\right )}{c}}}{2}\right )\right )}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {\frac {c}{-b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {1-\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}-\frac {4 c^{2} x^{4}}{\left (b -\sqrt {-4 a c +b^{2}}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right )}}\, \left (-\frac {2 c}{b +\sqrt {-4 a c +b^{2}}}+\frac {2 c}{b -\sqrt {-4 a c +b^{2}}}-\frac {b}{a}\right )}\right )}{2 \sqrt {\frac {-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}\) | \(809\) |
Maple 2021.1 output
\[ \int \frac {\sqrt {\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}+1}}{\sqrt {-\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}+1}}\, dx \]________________________________________________________________________________________