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