\[ \int \frac {f+g x}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-d g +e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{7 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{4}}-\frac {2 \left (-7 b e g +8 c d g +6 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{35 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right )^{3}}-\frac {8 c \left (-7 b e g +8 c d g +6 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{105 e^{2} \left (-b e +2 c d \right )^{3} \left (e x +d \right )^{2}}-\frac {16 c^{2} \left (-7 b e g +8 c d g +6 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{105 e^{2} \left (-b e +2 c d \right )^{4} \left (e x +d \right )} \]
command
integrate((g*x+f)/(e*x+d)^4/(-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 {2 \, {\left (8 \, c^{3} d^{4} g + 6 \, c^{3} d^{3} f e + 9 \, b c^{2} d^{3} g e - 56 \, {\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 - 42 \, {\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 - 63 \, {\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 - 168 \, {\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 + 12 \, b c^{2} d^{2} f e^{2} + 6 \, b^{2} c d^{2} g e^{2} - 126 \, {\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 + 21 \, {\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 d g e + 140 \, {\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 - 84 \, {\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} + 63 \, {\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} + 210 \, {\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 + 105 \, {\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 + 140 \, {\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 + 15 \, b^{2} c d f e^{3} - 15 \, b^{3} d g e^{3} - 252 \, {\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} - 21 \, {\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} - 105 \, {\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} + 15 \, b^{3} f e^{4}\right )} e^{\left (-2\right )}}{105 \, {\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 )}^{7}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________