\[ \int \frac {1}{\left (9+3 x-5 x^2+x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (3-x \right ) \left (1+x \right )}{8 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {3}{2}}}+\frac {5 \left (3-x \right )^{2} \left (1+x \right )}{64 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {3}{2}}}-\frac {15 \left (3-x \right )^{3} \left (1+x \right )}{256 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {3}{2}}}+\frac {15 \left (3-x \right )^{3} \left (1+x \right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {1+x}}{2}\right )}{512 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {3}{2}}} \]
command
integrate(1/(x^3-5*x^2+3*x+9)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {15 \, \log \left (\sqrt {x + 1} + 2\right )}{1024 \, \mathrm {sgn}\left (x - 3\right )} + \frac {15 \, \log \left ({\left | \sqrt {x + 1} - 2 \right |}\right )}{1024 \, \mathrm {sgn}\left (x - 3\right )} + \frac {1}{32 \, \sqrt {x + 1} \mathrm {sgn}\left (x - 3\right )} + \frac {7 \, {\left (x + 1\right )}^{\frac {3}{2}} - 36 \, \sqrt {x + 1}}{256 \, {\left (x - 3\right )}^{2} \mathrm {sgn}\left (x - 3\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________