\[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {3 b^{2} e}{\left (-a e +b d \right )^{4} \sqrt {\left (b x +a \right )^{2}}}-\frac {b^{2}}{2 \left (-a e +b d \right )^{3} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{2} \left (b x +a \right )}{2 \left (-a e +b d \right )^{3} \left (e x +d \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {3 b \,e^{2} \left (b x +a \right )}{\left (-a e +b d \right )^{4} \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {6 b^{2} e^{2} \left (b x +a \right ) \ln \left (b x +a \right )}{\left (-a e +b d \right )^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {6 b^{2} e^{2} \left (b x +a \right ) \ln \left (e x +d \right )}{\left (-a e +b d \right )^{5} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(1/(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
\[ \frac {6 \, b^{3} e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {6 \, b^{2} e^{3} \log \left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {12 \, b^{3} x^{3} e^{3} + 18 \, b^{3} d x^{2} e^{2} + 4 \, b^{3} d^{2} x e - b^{3} d^{3} + 18 \, a b^{2} x^{2} e^{3} + 28 \, a b^{2} d x e^{2} + 7 \, a b^{2} d^{2} e + 4 \, a^{2} b x e^{3} + 7 \, a^{2} b d e^{2} - a^{3} e^{3}}{2 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} {\left (b x^{2} e + b d x + a x e + a d\right )}^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________