\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (-c d x +a e \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{2 x^{2}}+\frac {3 \left (5 a^{2} e^{4}+10 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right ) \sqrt {c}\, \sqrt {d}}{8 \sqrt {e}}-\frac {3 \left (a^{2} e^{4}+10 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \arctanh \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}{2 \sqrt {a}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right ) \sqrt {a}\, \sqrt {e}}{8 \sqrt {d}}-\frac {3 \left (a e \left (a \,e^{2}+3 c \,d^{2}\right )-c d \left (3 a \,e^{2}+c \,d^{2}\right ) x \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 x} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^3/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{4} \, {\left (2 \, c^{2} d^{2} x e + \frac {{\left (5 \, c^{3} d^{4} e + 9 \, a c^{2} d^{2} e^{3}\right )} e^{\left (-1\right )}}{c d}\right )} \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} + \frac {3 \, {\left (5 \, a c^{2} d^{4} e + 10 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\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 )}{4 \, \sqrt {-a d e}} - \frac {3 \, {\left (\sqrt {c d} c^{3} d^{5} e^{\frac {1}{2}} + 10 \, \sqrt {c d} a c^{2} d^{3} e^{\frac {5}{2}} + 5 \, \sqrt {c d} a^{2} c d e^{\frac {9}{2}}\right )} e^{\left (-1\right )} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 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 )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right )}{8 \, c d} - \frac {7 \, {\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^{2} d^{5} e^{2} - 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 )}^{3} a c^{2} d^{4} e + 16 \, \sqrt {c d} a^{3} c d^{4} e^{\frac {7}{2}} - 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 )}^{2} \sqrt {c d} a^{2} c d^{3} e^{\frac {5}{2}} + 6 \, {\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 d^{3} e^{4} - 18 \, {\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 d^{2} e^{3} + 8 \, \sqrt {c d} a^{4} d^{2} e^{\frac {11}{2}} - 16 \, {\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} d e^{\frac {9}{2}} + 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^{4} d e^{6} - 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 )}^{3} a^{3} e^{5}}{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}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________