\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx \]
Optimal antiderivative \[ -\frac {g \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{2 c \,e^{2} \left (e x +d \right )}+\frac {\left (-b e +2 c d \right ) \left (-b e g -2 c d g +4 c e f \right ) \arctan \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right )}{8 c^{\frac {3}{2}} e^{2}}+\frac {\left (-b e g -2 c d g +4 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{4 c \,e^{2}} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{4} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, g x e^{\left (-1\right )} - \frac {{\left (4 \, c d g e - 4 \, c f e^{2} - b g e^{2}\right )} e^{\left (-3\right )}}{c}\right )} - \frac {{\left (4 \, c^{2} d^{2} g - 8 \, c^{2} d f e + 4 \, b c f e^{2} - b^{2} g e^{2}\right )} \sqrt {-c} e^{\left (-2\right )} \log \left ({\left | -b \sqrt {-c} e - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c \right |}\right )}{8 \, c^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________