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