\[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (15 a^{3} e^{6}+35 a^{2} c \,d^{2} e^{4}+45 a \,c^{2} d^{4} e^{2}+33 c^{3} d^{6}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{2048 c^{4} d^{4} e^{5}}+\frac {\left (\frac {5 a}{c d}-\frac {11 d}{e^{2}}\right ) x^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{112}+\frac {x^{3} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{8 e}-\frac {\left (231 c^{3} d^{6}-15 a \,c^{2} d^{4} e^{2}-95 a^{2} c \,d^{2} e^{4}-105 a^{3} e^{6}-10 c d e \left (-15 a^{2} e^{4}-10 a c \,d^{2} e^{2}+33 c^{2} d^{4}\right ) x \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{4480 c^{3} d^{3} e^{4}}+\frac {3 \left (-a \,e^{2}+c \,d^{2}\right )^{5} \left (15 a^{3} e^{6}+35 a^{2} c \,d^{2} e^{4}+45 a \,c^{2} d^{4} e^{2}+33 c^{3} d^{6}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{32768 c^{\frac {11}{2}} d^{\frac {11}{2}} e^{\frac {13}{2}}}-\frac {3 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (15 a^{3} e^{6}+35 a^{2} c \,d^{2} e^{4}+45 a \,c^{2} d^{4} e^{2}+33 c^{3} d^{6}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{16384 c^{5} d^{5} e^{6}} \]
command
integrate(x^3*(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}{573440} \, \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 \, {\left (4 \, {\left (14 \, c^{2} d^{2} x e + \frac {{\left (17 \, c^{9} d^{10} e^{7} + 33 \, a c^{8} d^{8} e^{9}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (c^{9} d^{11} e^{6} + 166 \, a c^{8} d^{9} e^{8} + 81 \, a^{2} c^{7} d^{7} e^{10}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x - \frac {{\left (11 \, c^{9} d^{12} e^{5} - 35 \, a c^{8} d^{10} e^{7} - 1075 \, a^{2} c^{7} d^{8} e^{9} - 5 \, a^{3} c^{6} d^{6} e^{11}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (99 \, c^{9} d^{13} e^{4} - 316 \, a c^{8} d^{11} e^{6} + 290 \, a^{2} c^{7} d^{9} e^{8} + 100 \, a^{3} c^{6} d^{7} e^{10} - 45 \, a^{4} c^{5} d^{5} e^{12}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x - \frac {{\left (231 \, c^{9} d^{14} e^{3} - 741 \, a c^{8} d^{12} e^{5} + 686 \, a^{2} c^{7} d^{10} e^{7} - 50 \, a^{3} c^{6} d^{8} e^{9} + 235 \, a^{4} c^{5} d^{6} e^{11} - 105 \, a^{5} c^{4} d^{4} e^{13}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (1155 \, c^{9} d^{15} e^{2} - 3738 \, a c^{8} d^{13} e^{4} + 3517 \, a^{2} c^{7} d^{11} e^{6} - 300 \, a^{3} c^{6} d^{9} e^{8} - 275 \, a^{4} c^{5} d^{7} e^{10} + 1190 \, a^{5} c^{4} d^{5} e^{12} - 525 \, a^{6} c^{3} d^{3} e^{14}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x - \frac {{\left (3465 \, c^{9} d^{16} e - 11445 \, a c^{8} d^{14} e^{3} + 11193 \, a^{2} c^{7} d^{12} e^{5} - 1325 \, a^{3} c^{6} d^{10} e^{7} - 925 \, a^{4} c^{5} d^{8} e^{9} - 1015 \, a^{5} c^{4} d^{6} e^{11} + 3675 \, a^{6} c^{3} d^{4} e^{13} - 1575 \, a^{7} c^{2} d^{2} e^{15}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} - \frac {3 \, {\left (33 \, c^{8} d^{16} - 120 \, a c^{7} d^{14} e^{2} + 140 \, a^{2} c^{6} d^{12} e^{4} - 40 \, a^{3} c^{5} d^{10} e^{6} - 10 \, a^{4} c^{4} d^{8} e^{8} - 8 \, a^{5} c^{3} d^{6} e^{10} - 20 \, a^{6} c^{2} d^{4} e^{12} + 40 \, a^{7} c d^{2} e^{14} - 15 \, a^{8} e^{16}\right )} e^{\left (-\frac {13}{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 )}{32768 \, \sqrt {c d} c^{5} d^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________