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