\[ \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {4 e^{3} \left (-a e +b d \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (-a e +b d \right )^{4}}{4 b^{5} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}-\frac {4 e \left (-a e +b d \right )^{3}}{3 b^{5} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {3 e^{2} \left (-a e +b d \right )^{2}}{b^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{4} \left (b x +a \right ) \ln \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((e*x+d)^4/(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
\[ \frac {e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {48 \, {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} x^{3} + 36 \, {\left (b^{3} d^{2} e^{2} + 2 \, a b^{2} d e^{3} - 3 \, a^{2} b e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{3} d^{3} e + 3 \, a b^{2} d^{2} e^{2} + 6 \, a^{2} b d e^{3} - 11 \, a^{3} e^{4}\right )} x + \frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4}}{b}}{12 \, {\left (b x + a\right )}^{4} b^{4} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________