\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^8 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (9 a^{2} e^{4}+10 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \left (2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{384 a^{3} d^{4} e^{3} x^{4}}-\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{7 d \,x^{7}}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^{2}}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{84 x^{6}}+\frac {\left (-63 a^{2} e^{4}+20 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{840 a^{2} d^{3} e^{2} x^{5}}-\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{5} \left (9 a^{2} e^{4}+10 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \arctanh \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}{2 \sqrt {a}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{2048 a^{\frac {9}{2}} d^{\frac {11}{2}} e^{\frac {9}{2}}}+\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (9 a^{2} e^{4}+10 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \left (2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{1024 a^{4} d^{5} e^{4} x^{2}} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^8/(e*x+d),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 \[ \text {Timed out} \]_______________________________________________________