\[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {4 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c^{2} d^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}-\frac {2 \left (e x +d \right )^{\frac {5}{2}}}{c d \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}} \]
command
integrate((e*x+d)^(7/2)/(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
\[ \frac {4 \, e^{2}}{3 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{2}} - \frac {2 \, {\left (c d^{2} e^{3} - a e^{5} + 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} e^{2}\right )}}{3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________