\[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (5 a^{2} e^{4}+10 a c \,d^{2} e^{2}+9 c^{2} d^{4}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{384 c^{3} d^{3} e^{4}}+\frac {x^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{7 e}+\frac {\left (63 c^{2} d^{4}-20 a c \,d^{2} e^{2}-35 a^{2} e^{4}-10 c d e \left (-5 a \,e^{2}+9 c \,d^{2}\right ) x \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{840 c^{2} d^{2} e^{3}}-\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{5} \left (5 a^{2} e^{4}+10 a c \,d^{2} e^{2}+9 c^{2} d^{4}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{2048 c^{\frac {9}{2}} d^{\frac {9}{2}} e^{\frac {11}{2}}}+\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (5 a^{2} e^{4}+10 a c \,d^{2} e^{2}+9 c^{2} d^{4}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{1024 c^{4} d^{4} e^{5}} \]
command
integrate(x^2*(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
\[ \frac {1}{107520} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, c^{2} d^{2} x e + \frac {{\left (15 \, c^{8} d^{9} e^{6} + 29 \, a c^{7} d^{7} e^{8}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (3 \, c^{8} d^{10} e^{5} + 380 \, a c^{7} d^{8} e^{7} + 185 \, a^{2} c^{6} d^{6} e^{9}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac {{\left (27 \, c^{8} d^{11} e^{4} - 93 \, a c^{7} d^{9} e^{6} - 2095 \, a^{2} c^{6} d^{7} e^{8} - 15 \, a^{3} c^{5} d^{5} e^{10}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (63 \, c^{8} d^{12} e^{3} - 218 \, a c^{7} d^{10} e^{5} + 228 \, a^{2} c^{6} d^{8} e^{7} + 90 \, a^{3} c^{5} d^{6} e^{9} - 35 \, a^{4} c^{4} d^{4} e^{11}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac {{\left (315 \, c^{8} d^{13} e^{2} - 1099 \, a c^{7} d^{11} e^{4} + 1166 \, a^{2} c^{6} d^{9} e^{6} - 150 \, a^{3} c^{5} d^{7} e^{8} + 455 \, a^{4} c^{4} d^{5} e^{10} - 175 \, a^{5} c^{3} d^{3} e^{12}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (945 \, c^{8} d^{14} e - 3360 \, a c^{7} d^{12} e^{3} + 3689 \, a^{2} c^{6} d^{10} e^{5} - 600 \, a^{3} c^{5} d^{8} e^{7} - 525 \, a^{4} c^{4} d^{6} e^{9} + 1400 \, a^{5} c^{3} d^{4} e^{11} - 525 \, a^{6} c^{2} d^{2} e^{13}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} + \frac {{\left (9 \, c^{7} d^{14} - 35 \, a c^{6} d^{12} e^{2} + 45 \, a^{2} c^{5} d^{10} e^{4} - 15 \, a^{3} c^{4} d^{8} e^{6} - 5 \, a^{4} c^{3} d^{6} e^{8} - 9 \, a^{5} c^{2} d^{4} e^{10} + 15 \, a^{6} c d^{2} e^{12} - 5 \, a^{7} e^{14}\right )} e^{\left (-\frac {11}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{2048 \, \sqrt {c d} c^{4} d^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________