\[ \int \frac {\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx \]
Optimal antiderivative \[ \frac {x \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}}{\sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}-\frac {\sqrt {\frac {1}{1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}\, \EllipticE \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}, \sqrt {-\frac {2 \sqrt {-4 a c +b^{2}}}{b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}\, \sqrt {b +\sqrt {-4 a c +b^{2}}}\, \sqrt {2}}{2 \sqrt {c}\, \sqrt {\frac {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}{1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}}+\frac {\sqrt {\frac {1}{1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}, \sqrt {-\frac {2 \sqrt {-4 a c +b^{2}}}{b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}\, \sqrt {b +\sqrt {-4 a c +b^{2}}}\, \sqrt {2}}{2 \sqrt {c}\, \sqrt {\frac {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}{1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}} \]
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 | \(\text {Expression too large to display}\) | \(1388\) |
Maple 2021.1 output
\[ \int \frac {\sqrt {\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}+1}}{\sqrt {\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}+1}}\, dx \]________________________________________________________________________________________