\[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{2 e \left (e x +d \right )^{\frac {7}{2}}}-\frac {3 c^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, \sqrt {2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}\right ) \sqrt {2}}{8 e \sqrt {d}}+\frac {3 c \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{4 e \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(9/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{8} \, {\left (\frac {3 \, \sqrt {2} c^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d}} + \frac {2 \, {\left (6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{3} d - 5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{2}\right )}}{{\left (x e + d\right )}^{2} c^{2}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________