\[ \int \frac {1}{\sqrt {a+b x^2-c x^4}} \, dx \]
Optimal antiderivative \[ \frac {\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {c}}{\sqrt {b +\sqrt {4 a c +b^{2}}}}, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{b -\sqrt {4 a c +b^{2}}}}\right ) \sqrt {1-\frac {2 c \,x^{2}}{b -\sqrt {4 a c +b^{2}}}}\, \sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {1-\frac {2 c \,x^{2}}{b +\sqrt {4 a c +b^{2}}}}\, \sqrt {2}}{2 \sqrt {c}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}} \]
command
integrate(1/(-c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {\frac {1}{2}} {\left (a^{\frac {3}{2}} \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + \sqrt {a} b\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}} {\rm ellipticF}\left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}}, -\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b^{2} + 2 \, a c}{2 \, a c}\right )}{2 \, a c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-c x^{4} + b x^{2} + a}}{c x^{4} - b x^{2} - a}, x\right ) \]