\[ \int \frac {1}{\left (1+\frac {2 x}{1+x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {3+\frac {3 x}{2}}{\sqrt {1+\frac {2 x}{x^{2}+1}}}+\frac {-x^{2}-1}{2 \left (1+x \right ) \sqrt {1+\frac {2 x}{x^{2}+1}}}-\frac {3 \left (1+x \right ) \arcsinh \left (x \right )}{\sqrt {x^{2}+1}\, \sqrt {1+\frac {2 x}{x^{2}+1}}}-\frac {9 \left (1+x \right ) \arctanh \left (\frac {\left (1-x \right ) \sqrt {2}}{2 \sqrt {x^{2}+1}}\right ) \sqrt {2}}{4 \sqrt {x^{2}+1}\, \sqrt {1+\frac {2 x}{x^{2}+1}}} \]
command
integrate(1/(1+2*x/(x^2+1))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {9 \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} - 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} - 2 \right |}}\right )}{4 \, \mathrm {sgn}\left (x + 1\right )} + \frac {3 \, \log \left (-x + \sqrt {x^{2} + 1}\right )}{\mathrm {sgn}\left (x + 1\right )} + \frac {\sqrt {x^{2} + 1}}{\mathrm {sgn}\left (x + 1\right )} + \frac {7 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{3} + 5 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 13 \, x + 13 \, \sqrt {x^{2} + 1} + 5}{{\left ({\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 2 \, x - 2 \, \sqrt {x^{2} + 1} - 1\right )}^{2} \mathrm {sgn}\left (x + 1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (\frac {2 \, x}{x^{2} + 1} + 1\right )}^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________