\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx \]
Optimal antiderivative \[ \frac {5 \left (-b e g -6 c d g +8 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{24 e^{2}}+\frac {\left (-b e g -6 c d g +8 c e f \right ) \left (-c e x -b e +c d \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{4 e^{2} \left (-b e +2 c d \right )}+\frac {2 \left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}}}{e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{3}}+\frac {5 \left (-b e +2 c d \right )^{3} \left (-b e g -6 c d g +8 c e f \right ) \arctan \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right )}{128 c^{\frac {3}{2}} e^{2}}+\frac {5 \left (-b e +2 c d \right ) \left (-b e g -6 c d g +8 c e f \right ) \left (2 c x +b \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{64 c e} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{192} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, {\left (4 \, {\left (6 \, c^{2} g x e - \frac {{\left (24 \, c^{5} d g e^{4} - 8 \, c^{5} f e^{5} - 17 \, b c^{4} g e^{5}\right )} e^{\left (-4\right )}}{c^{3}}\right )} x + \frac {{\left (180 \, c^{5} d^{2} g e^{3} - 144 \, c^{5} d f e^{4} - 204 \, b c^{4} d g e^{4} + 104 \, b c^{4} f e^{5} + 59 \, b^{2} c^{3} g e^{5}\right )} e^{\left (-4\right )}}{c^{3}}\right )} x - \frac {{\left (576 \, c^{5} d^{3} g e^{2} - 704 \, c^{5} d^{2} f e^{3} - 692 \, b c^{4} d^{2} g e^{3} + 848 \, b c^{4} d f e^{4} + 236 \, b^{2} c^{3} d g e^{4} - 264 \, b^{2} c^{3} f e^{5} - 15 \, b^{3} c^{2} g e^{5}\right )} e^{\left (-4\right )}}{c^{3}}\right )} - \frac {5 \, {\left (48 \, c^{4} d^{4} g - 64 \, c^{4} d^{3} f e - 64 \, b c^{3} d^{3} g e + 96 \, b c^{3} d^{2} f e^{2} + 24 \, b^{2} c^{2} d^{2} g e^{2} - 48 \, b^{2} c^{2} d f e^{3} + 8 \, b^{3} c f e^{4} - b^{4} g e^{4}\right )} \sqrt {-c} e^{\left (-2\right )} \log \left ({\left | -b \sqrt {-c} e - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c \right |}\right )}{128 \, c^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________