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