\[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {2}{3 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 c^{\frac {3}{2}} d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{\frac {5}{2}}}+\frac {2 c d}{\left (-a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e x +d}} \]
command
integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, c^{2} d^{2} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )}}{3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________