\[ \int \frac {1}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {\frac {1}{3 d \,x^{2}+9}}\, \sqrt {3 d \,x^{2}+9}\, \EllipticF \left (\frac {x \sqrt {d}\, \sqrt {3}}{\sqrt {3 d \,x^{2}+9}}, \frac {\sqrt {4-\frac {6 b}{d}}}{2}\right ) \sqrt {2}\, \sqrt {b \,x^{2}+2}}{2 \sqrt {d}\, \sqrt {\frac {b \,x^{2}+2}{d \,x^{2}+3}}\, \sqrt {d \,x^{2}+3}} \]
command
integrate(1/(b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {\sqrt {6} \sqrt {2} \sqrt {-b} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} \sqrt {-b} x, \frac {2 \, d}{3 \, b}\right )}{6 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}}{b d x^{4} + {\left (3 \, b + 2 \, d\right )} x^{2} + 6}, x\right ) \]