\[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx \]
Optimal antiderivative \[ \frac {35 b^{4} c^{2} \arctanh \left (\frac {\sqrt {a +b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{\frac {9}{2}}}-\frac {7 b \,c^{2} \sqrt {a +b \sqrt {\frac {c}{x}}}}{12 a^{2} \left (\frac {c}{x}\right )^{\frac {3}{2}}}+\frac {35 b^{2} c x \sqrt {a +b \sqrt {\frac {c}{x}}}}{48 a^{3}}+\frac {x^{2} \sqrt {a +b \sqrt {\frac {c}{x}}}}{2 a}-\frac {35 b^{3} c^{2} \sqrt {a +b \sqrt {\frac {c}{x}}}}{32 a^{4} \sqrt {\frac {c}{x}}} \]
command
integrate(x/(a+b*(c/x)^(1/2))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {105 \, b^{4} c^{4} \log \left (c^{2} {\left | b \right |}\right )}{\sqrt {a c} a^{4}} - \frac {105 \, b^{4} c^{4} \log \left ({\left | -b c^{2} - 2 \, \sqrt {a c} {\left (\sqrt {a c} \sqrt {c x} - \sqrt {a c^{2} x + \sqrt {c x} b c^{2}}\right )} \right |}\right )}{\sqrt {a c} a^{4}} - 2 \, \sqrt {a c^{2} x + \sqrt {c x} b c^{2}} {\left (2 \, \sqrt {c x} {\left (4 \, \sqrt {c x} {\left (\frac {7 \, b}{a^{2}} - \frac {6 \, \sqrt {c x}}{a c}\right )} - \frac {35 \, b^{2} c}{a^{3}}\right )} + \frac {105 \, b^{3} c^{2}}{a^{4}}\right )}}{192 \, c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x}{\sqrt {b \sqrt {\frac {c}{x}} + a}}\,{d x} \]________________________________________________________________________________________