\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {\left (\frac {a}{c d}-\frac {d}{e^{2}}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{16}+\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{5 e}-\frac {3 \left (-a \,e^{2}+c \,d^{2}\right )^{5} \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{256 c^{\frac {5}{2}} d^{\frac {5}{2}} e^{\frac {7}{2}}}+\frac {3 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{128 c^{2} d^{2} e^{3}} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{640} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, c^{2} d^{2} x e + \frac {{\left (11 \, c^{6} d^{7} e^{4} + 21 \, a c^{5} d^{5} e^{6}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (c^{6} d^{8} e^{3} + 64 \, a c^{5} d^{6} e^{5} + 31 \, a^{2} c^{4} d^{4} e^{7}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x - \frac {{\left (5 \, c^{6} d^{9} e^{2} - 23 \, a c^{5} d^{7} e^{4} - 233 \, a^{2} c^{4} d^{5} e^{6} - 5 \, a^{3} c^{3} d^{3} e^{8}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (15 \, c^{6} d^{10} e - 70 \, a c^{5} d^{8} e^{3} + 128 \, a^{2} c^{4} d^{6} e^{5} + 70 \, a^{3} c^{3} d^{4} e^{7} - 15 \, a^{4} c^{2} d^{2} e^{9}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} + \frac {3 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{256 \, \sqrt {c d} c^{2} d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________