\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5 (d+e x)} \, dx \]
Optimal antiderivative \[ \frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (35 a^{3} e^{6}+15 a^{2} c \,d^{2} e^{4}+9 a \,c^{2} d^{4} e^{2}+5 c^{3} d^{6}\right ) \arctanh \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}{2 \sqrt {a}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{128 a^{\frac {7}{2}} d^{\frac {9}{2}} e^{\frac {7}{2}}}-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 d \,x^{4}}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^{2}}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{24 x^{3}}+\frac {\left (-35 a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{96 a^{2} d^{3} e^{2} x^{2}}-\frac {\left (-105 a^{3} e^{6}+25 a^{2} c \,d^{2} e^{4}+17 a \,c^{2} d^{4} e^{2}+15 c^{3} d^{6}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{192 a^{3} d^{4} e^{3} x} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________