\[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-d g +e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{5 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{3}}-\frac {2 \left (-5 b e g +6 c d g +4 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{15 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right )^{2}}-\frac {4 c \left (-5 b e g +6 c d g +4 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{15 e^{2} \left (-b e +2 c d \right )^{3} \left (e x +d \right )} \]
command
integrate((g*x+f)/(e*x+d)^3/(-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
\[ \frac {2 \, {\left (3 \, c^{2} d^{3} g + 2 \, c^{2} d^{2} f e + 2 \, b c d^{2} g e - 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} c d^{2} g - 10 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} c d f e + 5 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b \sqrt {-c} d g e - 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} c d g + 3 \, b c d f e^{2} - 3 \, b^{2} d g e^{2} - 20 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} c f e - 5 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} b g e + 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{3} \sqrt {-c} g - 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b \sqrt {-c} f e^{2} + 3 \, b^{2} f e^{3}\right )} e^{\left (-2\right )}}{15 \, {\left (\sqrt {-c} d + \sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________