\[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3 c^{3} d^{3}}+\frac {2 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 c^{2} d^{2}}+\frac {2 \left (e x +d \right )^{\frac {7}{2}}}{7 c d}-\frac {2 \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{c^{\frac {9}{2}} d^{\frac {9}{2}}}+\frac {2 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {e x +d}}{c^{4} d^{4}} \]
command
integrate((e*x+d)^(9/2)/(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
\[ \frac {2 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{4} d^{4}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{6} d^{6} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{6} d^{7} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{6} d^{8} + 105 \, \sqrt {x e + d} c^{6} d^{9} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{5} d^{5} e^{2} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{5} d^{6} e^{2} - 315 \, \sqrt {x e + d} a c^{5} d^{7} e^{2} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c^{4} d^{4} e^{4} + 315 \, \sqrt {x e + d} a^{2} c^{4} d^{5} e^{4} - 105 \, \sqrt {x e + d} a^{3} c^{3} d^{3} e^{6}\right )}}{105 \, c^{7} d^{7}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________