\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (4 a d e +3 \left (a \,e^{2}+3 c \,d^{2}\right ) x \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{12 d \,x^{3}}-\frac {\left (-a^{3} e^{6}+15 a^{2} c \,d^{2} e^{4}+45 a \,c^{2} d^{4} e^{2}+5 c^{3} d^{6}\right ) \arctanh \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}{2 \sqrt {a}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{16 d^{\frac {3}{2}} \sqrt {a}\, \sqrt {e}}+\frac {c^{\frac {3}{2}} d^{\frac {3}{2}} \left (5 a \,e^{2}+3 c \,d^{2}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right ) \sqrt {e}}{2}-\frac {\left (5 c^{2} d^{4}+12 a c \,d^{2} e^{2}-a^{2} e^{4}-2 c d e \left (a \,e^{2}+7 c \,d^{2}\right ) x \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{8 d x} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^4/(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 \[ \text {Exception raised: TypeError} \]_______________________________________________________