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