\[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {e^{2} \left (-A e +B d \right )}{\left (-a e +b d \right )^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {-A b +a B}{4 b \left (-a e +b d \right ) \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {A e -B d}{3 \left (-a e +b d \right )^{2} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {e \left (-A e +B d \right )}{2 \left (-a e +b d \right )^{3} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {e^{3} \left (-A e +B d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{\left (-a e +b d \right )^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{3} \left (-A e +B d \right ) \left (b x +a \right ) \ln \left (e x +d \right )}{\left (-a e +b d \right )^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((B*x+A)/(e*x+d)/(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
\[ -\frac {{\left (B b d e^{3} - A b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {{\left (B d e^{4} - A e^{5}\right )} \log \left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {B a b^{4} d^{4} + 3 \, A b^{5} d^{4} - 6 \, B a^{2} b^{3} d^{3} e - 16 \, A a b^{4} d^{3} e + 18 \, B a^{3} b^{2} d^{2} e^{2} + 36 \, A a^{2} b^{3} d^{2} e^{2} - 10 \, B a^{4} b d e^{3} - 48 \, A a^{3} b^{2} d e^{3} - 3 \, B a^{5} e^{4} + 25 \, A a^{4} b e^{4} + 12 \, {\left (B b^{5} d^{2} e^{2} - B a b^{4} d e^{3} - A b^{5} d e^{3} + A a b^{4} e^{4}\right )} x^{3} - 6 \, {\left (B b^{5} d^{3} e - 8 \, B a b^{4} d^{2} e^{2} - A b^{5} d^{2} e^{2} + 7 \, B a^{2} b^{3} d e^{3} + 8 \, A a b^{4} d e^{3} - 7 \, A a^{2} b^{3} e^{4}\right )} x^{2} + 4 \, {\left (B b^{5} d^{4} - 6 \, B a b^{4} d^{3} e - A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} + 6 \, A a b^{4} d^{2} e^{2} - 13 \, B a^{3} b^{2} d e^{3} - 18 \, A a^{2} b^{3} d e^{3} + 13 \, A a^{3} b^{2} e^{4}\right )} x}{12 \, {\left (b d - a e\right )}^{5} {\left (b x + a\right )}^{4} b \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________