\[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {8 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c^{2} d^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}}}{3 c d \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {16 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e x +d}}{3 c^{3} d^{3} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}} \]
command
integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}}{\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{3}} + \frac {16 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}}{3 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{3}} + \frac {2 \, {\left (6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{7} d^{8} e^{3} - 6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{6} d^{6} e^{5} + {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{6} d^{6} e^{2}\right )} e^{\left (-3\right )}}{3 \, c^{9} d^{9}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________