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