\[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 b \,e^{2} \left (-5 A b e +2 a B e +3 B b d \right )}{\left (-a e +b d \right )^{6} \sqrt {\left (b x +a \right )^{2}}}-\frac {b \left (A b -a B \right )}{4 \left (-a e +b d \right )^{3} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {b \left (-3 A b e +2 a B e +B b d \right )}{3 \left (-a e +b d \right )^{4} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {3 b e \left (-2 A b e +a B e +B b d \right )}{2 \left (-a e +b d \right )^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {e^{3} \left (-A e +B d \right ) \left (b x +a \right )}{2 \left (-a e +b d \right )^{5} \left (e x +d \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {e^{3} \left (-5 A b e +a B e +4 B b d \right ) \left (b x +a \right )}{\left (-a e +b d \right )^{6} \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {5 b \,e^{3} \left (-3 A b e +a B e +2 B b d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{\left (-a e +b d \right )^{7} \sqrt {\left (b x +a \right )^{2}}}+\frac {5 b \,e^{3} \left (-3 A b e +a B e +2 B b d \right ) \left (b x +a \right ) \ln \left (e x +d \right )}{\left (-a e +b d \right )^{7} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \mathit {sage}_{0} x \]_______________________________________________________