\[ \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {6 b^{2} e^{2}}{\left (-a e +b d \right )^{5} \sqrt {\left (b x +a \right )^{2}}}-\frac {b^{2}}{3 \left (-a e +b d \right )^{3} \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}}}+\frac {3 b^{2} e}{2 \left (-a e +b d \right )^{4} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {e^{3} \left (b x +a \right )}{2 \left (-a e +b d \right )^{4} \left (e x +d \right )^{2} \sqrt {\left (b x +a \right )^{2}}}-\frac {4 b \,e^{3} \left (b x +a \right )}{\left (-a e +b d \right )^{5} \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}-\frac {10 b^{2} e^{3} \left (b x +a \right ) \ln \left (b x +a \right )}{\left (-a e +b d \right )^{6} \sqrt {\left (b x +a \right )^{2}}}+\frac {10 b^{2} e^{3} \left (b x +a \right ) \ln \left (e x +d \right )}{\left (-a e +b d \right )^{6} \sqrt {\left (b x +a \right )^{2}}} \]
command
integrate((b*x+a)/(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
\[ -\frac {10 \, b^{3} e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} b e^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {10 \, b^{2} e^{4} \log \left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{6} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{7} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, b^{5} d^{5} - 15 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 30 \, a^{4} b d e^{4} + 3 \, a^{5} e^{5} + 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 30 \, {\left (3 \, b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 5 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} d^{3} e^{2} + 21 \, a b^{4} d^{2} e^{3} - 12 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 24 \, a^{2} b^{3} d^{2} e^{3} + 32 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x}{6 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{3} {\left (x e + d\right )}^{2} \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{3}}\,{d x} \]________________________________________________________________________________________