\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{13 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{8}}-\frac {2 \left (-13 b e g +16 c d g +10 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{143 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right )^{7}}+\frac {16 c \left (13 b e g -2 c \left (8 d g +5 e f \right )\right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{1287 e^{2} \left (-b e +2 c d \right )^{3} \left (e x +d \right )^{6}}-\frac {32 c^{2} \left (-13 b e g +16 c d g +10 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{3003 e^{2} \left (-b e +2 c d \right )^{4} \left (e x +d \right )^{5}}+\frac {128 c^{3} \left (13 b e g -2 c \left (8 d g +5 e f \right )\right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{15015 e^{2} \left (-b e +2 c d \right )^{5} \left (e x +d \right )^{4}}-\frac {256 c^{4} \left (-13 b e g +16 c d g +10 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{45045 e^{2} \left (-b e +2 c d \right )^{6} \left (e x +d \right )^{3}} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,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 \[ \text {Timed out} \]_______________________________________________________