\[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {x^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{5 e}+\frac {\left (35 c^{2} d^{4}-12 a c \,d^{2} e^{2}-15 a^{2} e^{4}-6 c d e \left (-3 a \,e^{2}+7 c \,d^{2}\right ) x \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{240 c^{2} d^{2} e^{3}}+\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}+7 c^{2} d^{4}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{256 c^{\frac {7}{2}} d^{\frac {7}{2}} e^{\frac {9}{2}}}-\frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}+7 c^{2} d^{4}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{128 c^{3} d^{3} e^{4}} \]
command
integrate(x^2*(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
\[ \frac {1}{1920} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, c d x + \frac {{\left (c^{5} d^{6} e^{3} + 11 \, a c^{4} d^{4} e^{5}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x - \frac {{\left (7 \, c^{5} d^{7} e^{2} - 12 \, a c^{4} d^{5} e^{4} - 3 \, a^{2} c^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (35 \, c^{5} d^{8} e - 61 \, a c^{4} d^{6} e^{3} + 9 \, a^{2} c^{3} d^{4} e^{5} - 15 \, a^{3} c^{2} d^{2} e^{7}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x - \frac {{\left (105 \, c^{5} d^{9} - 190 \, a c^{4} d^{7} e^{2} + 36 \, a^{2} c^{3} d^{5} e^{4} + 30 \, a^{3} c^{2} d^{3} e^{6} - 45 \, a^{4} c d e^{8}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} - \frac {{\left (7 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 6 \, a^{2} c^{3} d^{6} e^{4} + 2 \, a^{3} c^{2} d^{4} e^{6} + 3 \, a^{4} c d^{2} e^{8} - 3 \, a^{5} e^{10}\right )} e^{\left (-\frac {9}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{256 \, \sqrt {c d} c^{3} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________