\[ \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {4 e \left (-a e +b d \right )^{3}}{b^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {\left (-a e +b d \right )^{4}}{2 b^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{3} \left (-3 a e +4 b d \right ) x \left (b x +a \right )}{b^{4} \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{4} x^{2} \left (b x +a \right )}{2 b^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {6 e^{2} \left (-a e +b d \right )^{2} \left (b x +a \right ) \ln \left (b x +a \right )}{b^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {6 \, {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {b^{3} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 8 \, b^{3} d x e^{3} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{2} x e^{4} \mathrm {sgn}\left (b x + a\right )}{2 \, b^{6}} - \frac {b^{4} d^{4} + 4 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} - 7 \, a^{4} e^{4} + 8 \, {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________