\[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {A e -B d}{\left (-a e +b d \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {-A b +a B}{2 b \left (-a e +b d \right ) \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {e \left (-A e +B d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{\left (-a e +b d \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {e \left (-A e +B d \right ) \left (b x +a \right ) \ln \left (e x +d \right )}{\left (-a e +b d \right )^{3} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)/(e*x+d)/(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
\[ -\frac {{\left (B b d e - A b e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )} + \frac {{\left (B d e^{2} - A e^{3}\right )} \log \left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {B a b^{2} d^{2} + A b^{3} d^{2} - 4 \, A a b^{2} d e - B a^{3} e^{2} + 3 \, A a^{2} b e^{2} + 2 \, {\left (B b^{3} d^{2} - B a b^{2} d e - A b^{3} d e + A a b^{2} e^{2}\right )} x}{2 \, {\left (b d - a e\right )}^{3} {\left (b x + a\right )}^{2} b \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________