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