\[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {d}\, \sqrt {a \,x^{2}+b}}{\sqrt {a}\, \sqrt {d \,x^{2}+c}}\right ) \sqrt {a \,x^{2}+b}}{x \sqrt {a}\, \sqrt {d}\, \sqrt {a +\frac {b}{x^{2}}}} \]
command
integrate(1/(a+b/x^2)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {a \log \left ({\left | -\sqrt {a d} \sqrt {b} + \sqrt {a^{2} c} \right |}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {a d} {\left | a \right |}} - \frac {a \log \left ({\left | -\sqrt {a x^{2} + b} \sqrt {a d} + \sqrt {a^{2} c + {\left (a x^{2} + b\right )} a d - a b d} \right |}\right )}{\sqrt {a d} {\left | a \right |} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________