\[ \int \frac {-b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{-b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}}} \, dx \]
Optimal antiderivative \[ -\frac {\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}, \sqrt {\frac {b -\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {b -\sqrt {-4 a c +b^{2}}}\, \sqrt {2}}{2 \sqrt {c}} \]
command
int((2*c*x^2-(-4*a*c+b^2)^(1/2)-b)/(1+2*c*x^2/(-b-(-4*a*c+b^2)^(1/2)))^(1/2)/(1+2*c/(-b+(-4*a*c+b^2)^(1/2))*x^2)^(1/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2538\) |
Maple 2021.1 output
\[ \int \frac {2 c \,x^{2}-b -\sqrt {-4 a c +b^{2}}}{\sqrt {\frac {2 c \,x^{2}}{-b -\sqrt {-4 a c +b^{2}}}+1}\, \sqrt {\frac {2 c \,x^{2}}{-b +\sqrt {-4 a c +b^{2}}}+1}}\, dx \]________________________________________________________________________________________