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