\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (b e g -6 c d g +4 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{2 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )}-\frac {2 \left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{3}}-\frac {3 \left (-b e +2 c d \right ) \left (b e g -6 c d g +4 c e f \right ) \arctan \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right )}{8 e^{2} \sqrt {c}}-\frac {3 \left (b e g -6 c d g +4 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{4 e^{2}} \]
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)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3 \, {\left (12 \, c^{2} d^{2} g - 8 \, c^{2} d f e - 8 \, b c d g e + 4 \, b c f e^{2} + b^{2} g e^{2}\right )} e^{\left (-2\right )} \log \left ({\left | -b e + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} \right |}\right )}{8 \, \sqrt {-c}} - \frac {1}{4} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, c g x e^{\left (-1\right )} - \frac {{\left (12 \, c^{2} d g e^{2} - 4 \, c^{2} f e^{3} - 5 \, b c g e^{3}\right )} e^{\left (-4\right )}}{c}\right )} - \frac {2 \, {\left (4 \, c^{2} d^{3} g - 4 \, c^{2} d^{2} f e - 4 \, b c d^{2} g e + 4 \, b c d f e^{2} + b^{2} d g e^{2} - b^{2} f e^{3}\right )} e^{\left (-2\right )}}{\sqrt {-c} d + \sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________