\[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {315 c^{4} d^{4} \arctan \left (\frac {\sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{\sqrt {-a \,e^{2}+c \,d^{2}}\, \sqrt {e x +d}}\right ) \sqrt {e}}{64 \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {11}{2}}}+\frac {1}{4 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {3 c d}{8 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {21 c^{2} d^{2}}{32 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {105 c^{3} d^{3}}{64 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {315 c^{4} d^{4} \sqrt {e x +d}}{64 \left (-a \,e^{2}+c \,d^{2}\right )^{5} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}} \]
command
integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {315 \, c^{4} d^{4} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{64 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {2 \, c^{4} d^{4} e}{{\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}} - \frac {{\left (325 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{7} d^{10} e^{4} - 975 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{6} d^{8} e^{6} + 765 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{6} d^{8} e^{3} + 975 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{5} d^{6} e^{8} - 1530 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{5} d^{6} e^{5} + 643 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{5} d^{6} e^{2} - 325 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{4} d^{4} e^{10} + 765 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{4} d^{4} e^{7} - 643 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{4} d^{4} e^{4} + 187 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{4} d^{4} e\right )} e^{\left (-4\right )}}{64 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} {\left (x e + d\right )}^{4} c^{4} d^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________