\[ \int x^{3/2} \coth ^{-1}\left (\sqrt {x}\right ) \, dx \]
Optimal antiderivative \[ \frac {x}{5}+\frac {x^{2}}{10}+\frac {2 x^{\frac {5}{2}} \mathrm {arccoth}\left (\sqrt {x}\right )}{5}+\frac {\ln \left (1-x \right )}{5} \]
command
integrate(x^(3/2)*arccoth(x^(1/2)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {8 \, {\left (\frac {{\left (\sqrt {x} + 1\right )}^{3}}{{\left (\sqrt {x} - 1\right )}^{3}} - \frac {{\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + \frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}}{5 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{4}} + \frac {2 \, {\left (\frac {5 \, {\left (\sqrt {x} + 1\right )}^{4}}{{\left (\sqrt {x} - 1\right )}^{4}} + \frac {10 \, {\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + 1\right )} \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{5 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{5}} + \frac {2}{5} \, \log \left (\frac {\sqrt {x} + 1}{{\left | \sqrt {x} - 1 \right |}}\right ) - \frac {2}{5} \, \log \left ({\left | \frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1 \right |}\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int x^{\frac {3}{2}} \operatorname {arcoth}\left (\sqrt {x}\right )\,{d x} \]________________________________________________________________________________________