\[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2}{7 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {4 c d}{7 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}-\frac {32 c^{2} d^{2} \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{21 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {256 c^{3} d^{3} e \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{21 \left (-a \,e^{2}+c \,d^{2}\right )^{6} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}} \]
command
integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \mathit {sage}_{0} x \]_______________________________________________________