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