\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx \]
Optimal antiderivative \[ \frac {\left (-b e +2 c d \right ) \left (8 c^{2} d^{2}+3 b^{2} e^{2}-4 c e \left (a e +2 b d \right )\right ) \left (b d -2 a e +\left (-b e +2 c d \right ) x \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{128 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{4} \left (e x +d \right )^{4}}-\frac {e \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{7 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{7}}-\frac {3 e \left (-b e +2 c d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{28 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{6}}-\frac {e \left (68 c^{2} d^{2}+21 b^{2} e^{2}-4 c e \left (4 a e +17 b d \right )\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{280 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{5}}+\frac {3 \left (-4 a c +b^{2}\right )^{2} \left (-b e +2 c d \right ) \left (8 c^{2} d^{2}+3 b^{2} e^{2}-4 c e \left (a e +2 b d \right )\right ) \arctanh \left (\frac {b d -2 a e +\left (-b e +2 c d \right ) x}{2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \sqrt {c \,x^{2}+b x +a}}\right )}{2048 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{\frac {11}{2}}}-\frac {3 \left (-4 a c +b^{2}\right ) \left (-b e +2 c d \right ) \left (8 c^{2} d^{2}+3 b^{2} e^{2}-4 c e \left (a e +2 b d \right )\right ) \left (b d -2 a e +\left (-b e +2 c d \right ) x \right ) \sqrt {c \,x^{2}+b x +a}}{1024 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{5} \left (e x +d \right )^{2}} \]
command
integrate((c*x^2+b*x+a)^(3/2)/(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 \[ \text {Timed out} \]_______________________________________________________