\[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {20 e^{3} \left (-a e +b d \right )^{3}}{b^{7} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (-a e +b d \right )^{6}}{4 b^{7} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {2 e \left (-a e +b d \right )^{5}}{b^{7} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {15 e^{2} \left (-a e +b d \right )^{4}}{2 b^{7} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{5} \left (-5 a e +6 b d \right ) x \left (b x +a \right )}{b^{6} \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{6} x^{2} \left (b x +a \right )}{2 b^{5} \sqrt {\left (b x +a \right )^{2}}}+\frac {15 e^{4} \left (-a e +b d \right )^{2} \left (b x +a \right ) \ln \left (b x +a \right )}{b^{7} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((e*x+d)^6/(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
\[ \frac {15 \, {\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {b^{5} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 12 \, b^{5} d x e^{5} \mathrm {sgn}\left (b x + a\right ) - 10 \, a b^{4} x e^{6} \mathrm {sgn}\left (b x + a\right )}{2 \, b^{10}} - \frac {b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} - 125 \, a^{4} b^{2} d^{2} e^{4} + 154 \, a^{5} b d e^{5} - 57 \, a^{6} e^{6} + 80 \, {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} - 18 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} - 7 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (2 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 20 \, a^{2} b^{4} d^{3} e^{3} - 110 \, a^{3} b^{3} d^{2} e^{4} + 130 \, a^{4} b^{2} d e^{5} - 47 \, a^{5} b e^{6}\right )} x}{4 \, {\left (b x + a\right )}^{4} b^{7} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________