\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (-c d x +3 a e \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{3 x}-\frac {\left (-5 a^{3} e^{6}-45 a^{2} c \,d^{2} e^{4}-15 a \,c^{2} d^{4} e^{2}+c^{3} d^{6}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{16 e^{\frac {3}{2}} \sqrt {c}\, \sqrt {d}}-\frac {a^{\frac {3}{2}} e^{\frac {3}{2}} \left (3 a \,e^{2}+5 c \,d^{2}\right ) \arctanh \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}{2 \sqrt {a}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right ) \sqrt {d}}{2}+\frac {\left (c^{2} d^{4}+28 a c \,d^{2} e^{2}+19 a^{2} e^{4}+2 c d e \left (7 a \,e^{2}+c \,d^{2}\right ) x \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{8 e} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^2/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{24} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, c^{2} d^{2} x e + \frac {{\left (7 \, c^{4} d^{5} e^{2} + 13 \, a c^{3} d^{3} e^{4}\right )} e^{\left (-2\right )}}{c^{2} d^{2}}\right )} x + \frac {{\left (3 \, c^{4} d^{6} e + 68 \, a c^{3} d^{4} e^{3} + 33 \, a^{2} c^{2} d^{2} e^{5}\right )} e^{\left (-2\right )}}{c^{2} d^{2}}\right )} + \frac {{\left (5 \, a^{2} c d^{3} e^{2} + 3 \, a^{3} d e^{4}\right )} \arctan \left (-\frac {\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}}{\sqrt {-a d e}}\right )}{\sqrt {-a d e}} + \frac {{\left (\sqrt {c d} c^{3} d^{6} e^{\frac {1}{2}} - 15 \, \sqrt {c d} a c^{2} d^{4} e^{\frac {5}{2}} - 45 \, \sqrt {c d} a^{2} c d^{2} e^{\frac {9}{2}} - 5 \, \sqrt {c d} a^{3} e^{\frac {13}{2}}\right )} e^{\left (-2\right )} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right )}{16 \, c d} - \frac {{\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{2} c d^{3} e^{2} + 2 \, \sqrt {c d} a^{3} d^{2} e^{\frac {7}{2}} + {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{3} d e^{4}}{a d e - {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________