\[ \int \frac {(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c^{2} d^{2}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}}}{5 c d}-\frac {2 \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{c^{\frac {7}{2}} d^{\frac {7}{2}}}+\frac {2 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e x +d}}{c^{3} d^{3}} \]
command
integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{3} d^{3}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{4} d^{4} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{5} + 15 \, \sqrt {x e + d} c^{4} d^{6} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{3} d^{3} e^{2} - 30 \, \sqrt {x e + d} a c^{3} d^{4} e^{2} + 15 \, \sqrt {x e + d} a^{2} c^{2} d^{2} e^{4}\right )}}{15 \, c^{5} d^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________