\[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {10 e^{3} \left (-a e +b d \right )^{2}}{b^{6} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (-a e +b d \right )^{5}}{4 b^{6} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {5 e \left (-a e +b d \right )^{4}}{3 b^{6} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {5 e^{2} \left (-a e +b d \right )^{3}}{b^{6} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{5} x \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {5 e^{4} \left (-a e +b d \right ) \left (b x +a \right ) \ln \left (b x +a \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((e*x+d)^5/(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
\[ \frac {x e^{5}}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {5 \, {\left (b d e^{4} - a e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {3 \, b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} - 125 \, a^{4} b d e^{4} + 77 \, a^{5} e^{5} + 120 \, {\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 60 \, {\left (b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} - 9 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 20 \, {\left (b^{5} d^{4} e + 2 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 22 \, a^{3} b^{2} d e^{4} + 13 \, a^{4} b e^{5}\right )} x}{12 \, {\left (b x + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________