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