\[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (5 a \,e^{2}+7 c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{192 c^{2} d^{2} e^{3}}-\frac {\left (\frac {5 a}{c d}+\frac {7 d}{e^{2}}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{60}+\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {7}{2}}}{6 c d e \left (e x +d \right )}+\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{5} \left (5 a \,e^{2}+7 c \,d^{2}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{1024 c^{\frac {7}{2}} d^{\frac {7}{2}} e^{\frac {9}{2}}}-\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (5 a \,e^{2}+7 c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{512 c^{3} d^{3} e^{4}} \]
command
integrate(x*(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
\[ \frac {1}{7680} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c^{2} d^{2} x e + \frac {{\left (13 \, c^{7} d^{8} e^{5} + 25 \, a c^{6} d^{6} e^{7}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (3 \, c^{7} d^{9} e^{4} + 278 \, a c^{6} d^{7} e^{6} + 135 \, a^{2} c^{5} d^{5} e^{8}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac {{\left (7 \, c^{7} d^{10} e^{3} - 27 \, a c^{6} d^{8} e^{5} - 423 \, a^{2} c^{5} d^{6} e^{7} - 5 \, a^{3} c^{4} d^{4} e^{9}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (35 \, c^{7} d^{11} e^{2} - 136 \, a c^{6} d^{9} e^{4} + 174 \, a^{2} c^{5} d^{7} e^{6} + 80 \, a^{3} c^{4} d^{5} e^{8} - 25 \, a^{4} c^{3} d^{3} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac {{\left (105 \, c^{7} d^{12} e - 415 \, a c^{6} d^{10} e^{3} + 546 \, a^{2} c^{5} d^{8} e^{5} - 150 \, a^{3} c^{4} d^{6} e^{7} + 245 \, a^{4} c^{3} d^{4} e^{9} - 75 \, a^{5} c^{2} d^{2} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} - \frac {{\left (7 \, c^{6} d^{12} - 30 \, a c^{5} d^{10} e^{2} + 45 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} - 15 \, a^{4} c^{2} d^{4} e^{8} + 18 \, a^{5} c d^{2} e^{10} - 5 \, a^{6} e^{12}\right )} e^{\left (-\frac {9}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{1024 \, \sqrt {c d} c^{3} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________