\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (6 a d e +\left (3 a \,e^{2}+11 c \,d^{2}\right ) x \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{24 d \,x^{4}}+\frac {\left (-3 a^{4} e^{8}+20 a^{3} c \,d^{2} e^{6}-90 a^{2} c^{2} d^{4} e^{4}-60 a \,c^{3} d^{6} e^{2}+5 c^{4} d^{8}\right ) \arctanh \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}{2 \sqrt {a}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{128 a^{\frac {3}{2}} d^{\frac {5}{2}} e^{\frac {3}{2}}}+c^{\frac {5}{2}} d^{\frac {5}{2}} e^{\frac {3}{2}} \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )-\frac {\left (2 a d e \left (-a \,e^{2}+5 c \,d^{2}\right ) \left (3 a \,e^{2}+c \,d^{2}\right )+\left (-3 a^{3} e^{6}+11 a^{2} c \,d^{2} e^{4}+83 a \,c^{2} d^{4} e^{2}+5 c^{3} d^{6}\right ) x \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{64 a \,d^{2} e \,x^{2}} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^5/(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 \[ \text {Exception raised: TypeError} \]_______________________________________________________