\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (5 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctanh \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}{2 \sqrt {a}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{16 a^{\frac {5}{2}} d^{\frac {7}{2}} e^{\frac {5}{2}}}-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{3 d \,x^{3}}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^{2}}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{12 x^{2}}+\frac {\left (-5 a \,e^{2}+3 c \,d^{2}\right ) \left (3 a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{24 a^{2} d^{3} e^{2} x} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\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 (-2\right )}}{8 \, \sqrt {-a d e} a^{2} d^{3}} - \frac {{\left (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 )} a^{2} c^{3} d^{8} e^{2} + 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 )}^{3} a c^{3} d^{7} e - 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 )}^{5} c^{3} d^{6} + 48 \, {\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^{2} c^{2} d^{6} e^{\frac {5}{2}} + 51 \, {\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} c^{2} d^{6} e^{4} + 72 \, {\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} c^{2} d^{5} e^{3} - 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 )}^{5} a c^{2} d^{4} e^{2} + 16 \, \sqrt {c d} a^{4} c d^{5} e^{\frac {11}{2}} + 144 \, {\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^{3} c d^{4} e^{\frac {9}{2}} + 105 \, {\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^{4} c d^{4} e^{6} + 24 \, {\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^{3} c d^{3} e^{5} - 9 \, {\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 )}^{5} a^{2} c d^{2} e^{4} + 48 \, \sqrt {c d} a^{5} d^{3} e^{\frac {15}{2}} + 33 \, {\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^{5} d^{2} e^{8} - 40 \, {\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^{4} d e^{7} + 15 \, {\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 )}^{5} a^{3} e^{6}\right )} e^{\left (-2\right )}}{24 \, {\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 )}^{3} a^{2} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________