\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {5 c \left (b e g -8 c d g +6 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2} \left (-b e +2 c d \right )}-\frac {2 \left (b e g -8 c d g +6 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{2}}-\frac {2 \left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}}}{e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{4}}-\frac {5 \left (-b e +2 c d \right )^{2} \left (b e g -8 c d g +6 c e f \right ) \arctan \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right )}{16 e^{2} \sqrt {c}}-\frac {5 \left (b e g -8 c d g +6 c e f \right ) \left (2 c x +b \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{8 e} \]
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)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {5 \, {\left (32 \, c^{3} d^{3} g - 24 \, c^{3} d^{2} f e - 36 \, b c^{2} d^{2} g e + 24 \, b c^{2} d f e^{2} + 12 \, b^{2} c d g e^{2} - 6 \, b^{2} c f e^{3} - b^{3} g e^{3}\right )} e^{\left (-2\right )} \log \left ({\left | -b e + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} \right |}\right )}{16 \, \sqrt {-c}} + \frac {1}{24} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, {\left (4 \, c^{2} g x - \frac {{\left (24 \, c^{4} d g e^{3} - 6 \, c^{4} f e^{4} - 13 \, b c^{3} g e^{4}\right )} e^{\left (-4\right )}}{c^{2}}\right )} x + \frac {{\left (184 \, c^{4} d^{2} g e^{2} - 96 \, c^{4} d f e^{3} - 160 \, b c^{3} d g e^{3} + 54 \, b c^{3} f e^{4} + 33 \, b^{2} c^{2} g e^{4}\right )} e^{\left (-4\right )}}{c^{2}}\right )} - \frac {2 \, {\left (8 \, c^{3} d^{4} g - 8 \, c^{3} d^{3} f e - 12 \, b c^{2} d^{3} g e + 12 \, b c^{2} d^{2} f e^{2} + 6 \, b^{2} c d^{2} g e^{2} - 6 \, b^{2} c d f e^{3} - b^{3} d g e^{3} + b^{3} f e^{4}\right )} e^{\left (-2\right )}}{\sqrt {-c} d + \sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________