\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{5 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{5}} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2}{15} \, {\left (\frac {3 \, \sqrt {c d} c^{2} d^{2} e^{\frac {1}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c d^{2} e^{4} - a e^{6}} - \frac {\frac {{\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}}\right )} c^{2} d^{4} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}} - \frac {10 \, {\left (3 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c d e - {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}}\right )} c^{2} d^{3} e \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c d^{2} e^{2} - a e^{4}} - \frac {2 \, {\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}}\right )} a c d^{2} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}} + 15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{2} d^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {10 \, {\left (3 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c d e - {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}}\right )} a c d e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c d^{2} e^{2} - a e^{4}} + \frac {{\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}}\right )} a^{2} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}}}{c d^{2} e^{4} - a e^{6}}\right )} e \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________