\[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {16 c \left (-7 b e g +4 c d g +10 c e f \right ) \left (2 c x +b \right )}{105 e \left (-b e +2 c d \right )^{4} \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}-\frac {2 \left (-d g +e f \right )}{7 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{2} \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}-\frac {2 \left (-7 b e g +4 c d g +10 c e f \right )}{35 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}+\frac {128 c^{2} \left (-7 b e g +4 c d g +10 c e f \right ) \left (2 c x +b \right )}{105 e \left (-b e +2 c d \right )^{6} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),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 \[ \mathit {sage}_{0} x \]_______________________________________________________