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