\[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {\left (\frac {a}{c d}-\frac {3 d}{e^{2}}\right ) x^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{20}+\frac {x^{3} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{6 e}-\frac {\left (105 c^{3} d^{6}-21 a \,c^{2} d^{4} e^{2}-33 a^{2} c \,d^{2} e^{4}-35 a^{3} e^{6}-6 c d e \left (-7 a^{2} e^{4}-6 a c \,d^{2} e^{2}+21 c^{2} d^{4}\right ) x \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{960 c^{3} d^{3} e^{4}}-\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (7 a^{3} e^{6}+15 a^{2} c \,d^{2} e^{4}+21 a \,c^{2} d^{4} e^{2}+21 c^{3} d^{6}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{1024 c^{\frac {9}{2}} d^{\frac {9}{2}} e^{\frac {11}{2}}}+\frac {\left (-7 a^{4} e^{8}-8 a^{3} c \,d^{2} e^{6}-6 a^{2} c^{2} d^{4} e^{4}+21 c^{4} d^{8}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{512 c^{4} d^{4} e^{5}} \]
command
integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{7680} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c d x + \frac {{\left (c^{6} d^{7} e^{4} + 13 \, a c^{5} d^{5} e^{6}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac {{\left (9 \, c^{6} d^{8} e^{3} - 14 \, a c^{5} d^{6} e^{5} - 3 \, a^{2} c^{4} d^{4} e^{7}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (21 \, c^{6} d^{9} e^{2} - 33 \, a c^{5} d^{7} e^{4} + 3 \, a^{2} c^{4} d^{5} e^{6} - 7 \, a^{3} c^{3} d^{3} e^{8}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac {{\left (105 \, c^{6} d^{10} e - 168 \, a c^{5} d^{8} e^{3} + 18 \, a^{2} c^{4} d^{6} e^{5} + 16 \, a^{3} c^{3} d^{4} e^{7} - 35 \, a^{4} c^{2} d^{2} e^{9}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (315 \, c^{6} d^{11} - 525 \, a c^{5} d^{9} e^{2} + 78 \, a^{2} c^{4} d^{7} e^{4} + 54 \, a^{3} c^{3} d^{5} e^{6} + 55 \, a^{4} c^{2} d^{3} e^{8} - 105 \, a^{5} c d e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} + \frac {{\left (21 \, c^{6} d^{12} - 42 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} + 4 \, a^{3} c^{3} d^{6} e^{6} + 3 \, a^{4} c^{2} d^{4} e^{8} + 6 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\right )} e^{\left (-\frac {11}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{1024 \, \sqrt {c d} c^{4} d^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________