\[ \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {1}{2 b \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(1/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{2 \, {\left (b x + a\right )}^{2} b \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________