\[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {e^{2}}{\left (-a e +b d \right )^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {1}{3 \left (-a e +b d \right ) \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {e}{2 \left (-a e +b d \right )^{2} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {e^{3} \left (b x +a \right ) \ln \left (b x +a \right )}{\left (-a e +b d \right )^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{3} \left (b x +a \right ) \ln \left (e x +d \right )}{\left (-a e +b d \right )^{4} \sqrt {\left (b x +a \right )^{2}}} \]
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
\[ -\frac {b e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right )} + \frac {e^{4} \log \left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x}{6 \, {\left (b d - a e\right )}^{4} {\left (b x + a\right )}^{3} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}}\,{d x} \]________________________________________________________________________________________