\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{5 d \,x^{5}}-\frac {\left (\frac {3 c}{a e}-\frac {7 e}{d^{2}}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{40 x^{4}}+\frac {\left (-35 a^{2} e^{4}+12 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{240 a^{2} d^{3} e^{2} x^{3}}+\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (7 a^{2} e^{4}+6 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right ) \arctanh \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}{2 \sqrt {a}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{256 a^{\frac {7}{2}} d^{\frac {9}{2}} e^{\frac {7}{2}}}-\frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (7 a^{2} e^{4}+6 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right ) \left (2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{128 a^{3} d^{4} e^{3} x^{2}} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^6/(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________