\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx \]
Optimal antiderivative \[ \frac {5 c \left (3 b e g -10 c d g +4 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{6 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )}+\frac {2 \left (3 b e g -10 c d g +4 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{3}}-\frac {2 \left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}}}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{5}}+\frac {5 \left (-b e +2 c d \right ) \left (3 b e g -10 c d g +4 c e f \right ) \arctan \left (\frac {e \left (2 c x +b \right )}{2 \sqrt {c}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}\right ) \sqrt {c}}{8 e^{2}}+\frac {5 c \left (3 b e g -10 c d g +4 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{4 e^{2}} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________