6.1 Problem number 1735

\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x} \, dx \]

Optimal antiderivative \[ \frac {2 \arctanh \! \left (\frac {\sqrt {a +\frac {b}{x}}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {2}{a \sqrt {a +\frac {b}{x}}} \]

command

integrate(1/(a+b/x)^(3/2)/x,x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \text {Exception raised: TypeError} \]

Giac 1.7.0 via sagemath 9.3 output

\[ -\frac {2 \, \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {2}{a \sqrt {\frac {a x + b}{x}}} \]