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