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