\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3 (d+e x)} \, dx \]
Optimal antiderivative \[ \frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (3 a \,e^{2}+c \,d^{2}\right ) \arctanh \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}{2 \sqrt {a}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{8 a^{\frac {3}{2}} d^{\frac {5}{2}} e^{\frac {3}{2}}}-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{2 d \,x^{2}}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^{2}}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 x} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^3/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \arctan \left (-\frac {\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}}{\sqrt {-a d e}}\right ) e^{\left (-1\right )}}{4 \, \sqrt {-a d e} a d^{2}} + \frac {{\left ({\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a c^{2} d^{5} e + {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} c^{2} d^{4} + 8 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2} \sqrt {c d} a c d^{3} e^{\frac {3}{2}} + 10 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{2} c d^{3} e^{3} + 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} a c d^{2} e^{2} + 8 \, \sqrt {c d} a^{3} d^{2} e^{\frac {9}{2}} + 5 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{3} d e^{5} - 3 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} a^{2} e^{4}\right )} e^{\left (-1\right )}}{4 \, {\left (a d e - {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2}\right )}^{2} a d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________