\[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {2}{7 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}+\frac {2 c d}{5 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 c^{2} d^{2}}{3 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 c^{\frac {7}{2}} d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{\frac {9}{2}}}+\frac {2 c^{3} d^{3}}{\left (-a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {e x +d}} \]
command
integrate(1/(e*x+d)^(7/2)/(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
\[ \frac {2 \, c^{4} d^{4} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} + \frac {2 \, {\left (105 \, {\left (x e + d\right )}^{3} c^{3} d^{3} + 35 \, {\left (x e + d\right )}^{2} c^{3} d^{4} + 21 \, {\left (x e + d\right )} c^{3} d^{5} + 15 \, c^{3} d^{6} - 35 \, {\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 42 \, {\left (x e + d\right )} a c^{2} d^{3} e^{2} - 45 \, a c^{2} d^{4} e^{2} + 21 \, {\left (x e + d\right )} a^{2} c d e^{4} + 45 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6}\right )}}{105 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left (x e + d\right )}^{\frac {7}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________