\[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {35 c^{3} d^{3} \arctan \left (\frac {\sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{\sqrt {-a \,e^{2}+c \,d^{2}}\, \sqrt {e x +d}}\right ) \sqrt {e}}{8 \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {9}{2}}}+\frac {1}{3 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {7 c d}{12 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {35 c^{2} d^{2}}{24 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {35 c^{3} d^{3} \sqrt {e x +d}}{8 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}} \]
command
integrate(1/(e*x+d)^(5/2)/(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
\[ -\frac {35 \, c^{3} d^{3} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{8 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {2 \, c^{3} d^{3} e}{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}} - \frac {{\left (87 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{5} d^{7} e^{3} - 174 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{4} d^{5} e^{5} + 136 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{5} e^{2} + 87 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{3} d^{3} e^{7} - 136 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{3} d^{3} e^{4} + 57 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{3} d^{3} e\right )} e^{\left (-3\right )}}{24 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left (x e + d\right )}^{3} c^{3} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________