\[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {d g -e f}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {3}{2}} \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}-\frac {35 c^{2} \left (-2 b e g +c d g +3 c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{8 e^{2} \left (-b e +2 c d \right )^{\frac {11}{2}}}+\frac {2 b e g -c d g -3 c e f}{4 e^{2} \left (-b e +2 c d \right )^{2} \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}} \sqrt {e x +d}}+\frac {7 c \left (-2 b e g +c d g +3 c e f \right ) \sqrt {e x +d}}{12 e^{2} \left (-b e +2 c d \right )^{3} \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}-\frac {35 c \left (-2 b e g +c d g +3 c e f \right )}{24 e^{2} \left (-b e +2 c d \right )^{4} \sqrt {e x +d}\, \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {35 c^{2} \left (-2 b e g +c d g +3 c e f \right ) \sqrt {e x +d}}{8 e^{2} \left (-b e +2 c d \right )^{5} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((g*x+f)/(e*x+d)^(3/2)/(-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 \[ \text {Timed out} \]_______________________________________________________