\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{5 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{4}}-\frac {2 \left (-5 b e g +8 c d g +2 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{15 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right )^{3}} \]
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)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (4 \, c^{3} d^{4} g + c^{3} d^{3} f e - 6 \, b c^{2} d^{3} g e - 5 \, {\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^{2} d^{3} g - 5 \, {\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^{2} d^{2} f e - 15 \, {\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} c d^{2} g e - 25 \, {\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^{2} d^{2} g + b c^{2} d^{2} f e^{2} + 8 \, b^{2} c d^{2} g e^{2} + 5 \, {\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^{2} d f e + 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{3} \sqrt {-c} c d g + 10 \, {\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} c d f e^{2} + 5 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b^{2} \sqrt {-c} d g e^{2} + 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{3} \sqrt {-c} c f e + 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{3} b \sqrt {-c} g e + 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{4} c g - 3 \, b^{2} c d f e^{3} - 3 \, b^{3} d g e^{3} - 25 \, {\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 c f e^{2} - 5 \, {\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^{2} g e^{2} - 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b^{2} \sqrt {-c} f e^{3} + 3 \, b^{3} f e^{4}\right )} e^{\left (-2\right )}}{15 \, {\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 )}^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________