\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {c}{x}}} x} \, dx \]
Optimal antiderivative \[ \frac {4 \arctanh \left (\frac {\sqrt {a +b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
command
integrate(1/x/(a+b*(c/x)^(1/2))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, \sqrt {c} {\left (\frac {\sqrt {a c} \log \left (\sqrt {{\left | a \right |} {\left | c \right |}} {\left | b \right |} {\left | c \right |}\right )}{a c} - \frac {\sqrt {a c} \log \left ({\left | -\sqrt {a c} b c - 2 \, {\left (\sqrt {a c} \sqrt {c x} - \sqrt {a c^{2} x + \sqrt {c x} b c^{2}}\right )} a \right |}\right )}{a c}\right )}}{\mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {b \sqrt {\frac {c}{x}} + a} x}\,{d x} \]________________________________________________________________________________________