\[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {b \left (-3 A b e +2 a B e +B b d \right )}{\left (-a e +b d \right )^{4} \sqrt {\left (b x +a \right )^{2}}}-\frac {b \left (A b -a B \right )}{2 \left (-a e +b d \right )^{3} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {e \left (-A e +B d \right ) \left (b x +a \right )}{2 \left (-a e +b d \right )^{3} \left (e x +d \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {e \left (-3 A b e +a B e +2 B b d \right ) \left (b x +a \right )}{\left (-a e +b d \right )^{4} \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {3 b e \left (-2 A b e +a B e +B b d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{\left (-a e +b d \right )^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {3 b e \left (-2 A b e +a B e +B b d \right ) \left (b x +a \right ) \ln \left (e x +d \right )}{\left (-a e +b d \right )^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)/(e*x+d)^3/(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
\[ -\frac {3 \, {\left (B b^{3} d e + B a b^{2} e^{2} - 2 \, A b^{3} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {3 \, {\left (B b^{2} d e^{2} + B a b e^{3} - 2 \, A b^{2} e^{3}\right )} \log \left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {6 \, B b^{3} d x^{3} e^{2} + 9 \, B b^{3} d^{2} x^{2} e + 2 \, B b^{3} d^{3} x + 6 \, B a b^{2} x^{3} e^{3} - 12 \, A b^{3} x^{3} e^{3} + 18 \, B a b^{2} d x^{2} e^{2} - 18 \, A b^{3} d x^{2} e^{2} + 16 \, B a b^{2} d^{2} x e - 4 \, A b^{3} d^{2} x e + B a b^{2} d^{3} + A b^{3} d^{3} + 9 \, B a^{2} b x^{2} e^{3} - 18 \, A a b^{2} x^{2} e^{3} + 16 \, B a^{2} b d x e^{2} - 28 \, A a b^{2} d x e^{2} + 10 \, B a^{2} b d^{2} e - 7 \, A a b^{2} d^{2} e + 2 \, B a^{3} x e^{3} - 4 \, A a^{2} b x e^{3} + B a^{3} d e^{2} - 7 \, A a^{2} b d e^{2} + A a^{3} e^{3}}{2 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} {\left (b x^{2} e + b d x + a x e + a d\right )}^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________