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