\[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {10 b^{2} e^{3}}{\left (-a e +b d \right )^{6} \sqrt {\left (b x +a \right )^{2}}}-\frac {b^{2}}{4 \left (-a e +b d \right )^{3} \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}}}+\frac {b^{2} e}{\left (-a e +b d \right )^{4} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {3 b^{2} e^{2}}{\left (-a e +b d \right )^{5} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {e^{4} \left (b x +a \right )}{2 \left (-a e +b d \right )^{5} \left (e x +d \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {5 b \,e^{4} \left (b x +a \right )}{\left (-a e +b d \right )^{6} \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}+\frac {15 b^{2} e^{4} \left (b x +a \right ) \ln \left (b x +a \right )}{\left (-a e +b d \right )^{7} \sqrt {\left (b x +a \right )^{2}}}-\frac {15 b^{2} e^{4} \left (b x +a \right ) \ln \left (e x +d \right )}{\left (-a e +b d \right )^{7} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate(1/(e*x+d)^3/(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 \, b^{3} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{8} d^{7} \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{7} d^{6} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{6} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{5} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{4} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{3} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b^{2} d e^{6} \mathrm {sgn}\left (b x + a\right ) - a^{7} b e^{7} \mathrm {sgn}\left (b x + a\right )} - \frac {15 \, b^{2} e^{5} \log \left ({\left | x e + d \right |}\right )}{b^{7} d^{7} e \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{6} d^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{5} d^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{4} d^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{3} d^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{2} d^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b d e^{7} \mathrm {sgn}\left (b x + a\right ) - a^{7} e^{8} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{6} d^{6} - 8 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} - 80 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 24 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 60 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} - 30 \, {\left (3 \, b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} - 7 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} + 15 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 13 \, a^{3} b^{3} e^{6}\right )} x^{3} + 5 \, {\left (b^{6} d^{4} e^{2} - 16 \, a b^{5} d^{3} e^{3} - 66 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 25 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (b^{6} d^{5} e - 10 \, a b^{5} d^{4} e^{2} + 60 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 95 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x}{4 \, {\left (b d - a e\right )}^{7} {\left (b x + a\right )}^{4} {\left (x e + d\right )}^{2} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________