\[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{3/2}} \, dx \]
Optimal antiderivative \[ -8 \arctan \! \left (\frac {\sqrt {x}}{\sqrt {x^{2}-x}}\right )+4 \arctan \! \left (2 \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}-\frac {2 \ln \! \left (-1+4 x +4 \sqrt {x^{2}-x}\right )}{\sqrt {x}}-\frac {4 \arctan \! \left (\frac {2 \sqrt {-1+x}\, \sqrt {2}}{3}\right ) \sqrt {x^{2}-x}\, \sqrt {2}}{\sqrt {-1+x}\, \sqrt {x}} \]
command
integrate(log(-1+4*x+4*((-1+x)*x)^(1/2))/x^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: NotImplementedError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ 2 \, \sqrt {2} \pi i - 4 \, \pi i + 4 \, \pi \mathrm {sgn}\left (\sqrt {x - 1} - \sqrt {x}\right ) - 2 \, \sqrt {2} {\left (\pi \mathrm {sgn}\left (\sqrt {x - 1} - \sqrt {x}\right ) + 2 \, \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1\right )}}{3 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} + 4 \, \sqrt {2} \arctan \left (\frac {2}{3} \, \sqrt {2} i\right ) + 4 \, \sqrt {2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) - \frac {2 \, \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right )}{\sqrt {x}} - 8 \, \arctan \left (i\right ) + 8 \, \arctan \left (\frac {{\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1}{2 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right ) \]