\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ \frac {2 \left (3 b e g -8 c d g +2 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{2}}-\frac {2 \left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{4}}+\frac {\left (3 b e g -8 c d g +2 c e f \right ) \arctan \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right ) \sqrt {c}}{2 e^{2}}+\frac {c \left (3 b e g -8 c d g +2 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{e^{2} \left (-b e +2 c d \right )} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} c g e^{\left (-2\right )} + \frac {{\left (8 \, c^{2} d g - 2 \, c^{2} f e - 3 \, b c g e\right )} e^{\left (-2\right )} \log \left ({\left | -b e + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} \right |}\right )}{2 \, \sqrt {-c}} - \frac {2 \, {\left (20 \, c^{3} d^{4} g - 8 \, c^{3} d^{3} f e - 18 \, b c^{2} d^{3} g e - 36 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} c^{2} d^{3} g + 12 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} c^{2} d^{2} f e + 24 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b \sqrt {-c} c d^{2} g e - 24 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} c^{2} d^{2} g + 6 \, b c^{2} d^{2} f e^{2} + 6 \, b^{2} c d^{2} g e^{2} + 12 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} c^{2} d f e + 18 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} b c d g e - 3 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b^{2} \sqrt {-c} d g e^{2} - 3 \, b^{2} c d f e^{3} - b^{3} d g e^{3} - 6 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} b c f e^{2} - 3 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} b^{2} g e^{2} - 3 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b^{2} \sqrt {-c} f e^{3} + b^{3} f e^{4}\right )} e^{\left (-2\right )}}{3 \, {\left (\sqrt {-c} d + \sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________