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