\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{7 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{5}}+\frac {8 c d \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{35 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{4}}+\frac {16 c^{2} d^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{105 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )^{3}} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2}{105} \, {\left (\frac {8 \, \sqrt {c d} c^{3} d^{3} e^{\frac {1}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}} + \frac {\frac {3 \, {\left (35 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{3} d^{3} e^{3} - 35 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} + 21 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} c d e - 5 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}}\right )} c d^{2} e \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}} - \frac {7 \, {\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}}\right )} c d \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}} - \frac {3 \, {\left (35 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{3} d^{3} e^{3} - 35 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} + 21 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} c d e - 5 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}}\right )} a e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}}}{c d^{2} e - a e^{3}}\right )} e \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________