\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {\frac {1}{1+\frac {d \,x^{2}}{c}}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (\frac {x \sqrt {d}}{\sqrt {c}\, \sqrt {1+\frac {d \,x^{2}}{c}}}, \sqrt {1-\frac {b c}{a d}}\right ) \sqrt {c}\, \sqrt {b \,x^{2}+a}}{a \sqrt {d}\, \sqrt {\frac {c \left (b \,x^{2}+a \right )}{a \left (d \,x^{2}+c \right )}}\, \sqrt {d \,x^{2}+c}} \]
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
\[ -\frac {\sqrt {a c} \sqrt {-\frac {b}{a}} {\rm ellipticF}\left (x \sqrt {-\frac {b}{a}}, \frac {a d}{b c}\right )}{b c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{b d x^{4} + {\left (b c + a d\right )} x^{2} + a c}, x\right ) \]