\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{3}}-\frac {g \arctan \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right ) \sqrt {c}}{e^{2}}-\frac {2 g \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{e^{2} \left (e x +d \right )} \]
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)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {c g e^{\left (-2\right )} \log \left ({\left | b e - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} \right |}\right )}{\sqrt {-c}} - \frac {2 \, {\left (7 \, c^{2} d^{3} g - c^{2} d^{2} f e - 4 \, b c d^{2} g e - 12 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} c d^{2} g + 3 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b \sqrt {-c} d g e - 9 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} c d g + b c d f e^{2} + b^{2} d g e^{2} + 3 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} c f e + 3 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} b g e + 3 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b \sqrt {-c} f e^{2} - b^{2} f e^{3}\right )} e^{\left (-2\right )}}{3 \, {\left (\sqrt {-c} d + \sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________