\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {5 c d \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{3 e^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{e \left (e x +d \right )^{\frac {7}{2}}}+\frac {5 c d \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{\sqrt {-a \,e^{2}+c \,d^{2}}\, \sqrt {e x +d}}\right )}{e^{\frac {7}{2}}}+\frac {5 c d \left (a -\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{e \sqrt {e x +d}} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(9/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (2 \, {\left (6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{4} e^{3} - 6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e^{5} - {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2}\right )} e^{\left (-3\right )} - \frac {15 \, {\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{\sqrt {c d^{2} e - a e^{3}}} + \frac {3 \, {\left (\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{6} e - 2 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{3} d^{4} e^{3} + \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{2} d^{2} e^{5}\right )} e^{\left (-1\right )}}{{\left (x e + d\right )} c d}\right )} e^{\left (-4\right )}}{3 \, c d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________