\[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx \]
Optimal antiderivative \[ -\frac {\left (\frac {3 a}{c d}+\frac {5 d}{e^{2}}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{24}+\frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{4 c d e \left (e x +d \right )}-\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (3 a \,e^{2}+5 c \,d^{2}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{128 c^{\frac {5}{2}} d^{\frac {5}{2}} e^{\frac {7}{2}}}+\frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (3 a \,e^{2}+5 c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{64 c^{2} d^{2} e^{3}} \]
command
integrate(x*(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
\[ \frac {1}{192} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (6 \, c d x + \frac {{\left (c^{4} d^{5} e^{2} + 9 \, a c^{3} d^{3} e^{4}\right )} e^{\left (-3\right )}}{c^{3} d^{3}}\right )} x - \frac {{\left (5 \, c^{4} d^{6} e - 10 \, a c^{3} d^{4} e^{3} - 3 \, a^{2} c^{2} d^{2} e^{5}\right )} e^{\left (-3\right )}}{c^{3} d^{3}}\right )} x + \frac {{\left (15 \, c^{4} d^{7} - 31 \, a c^{3} d^{5} e^{2} + 9 \, a^{2} c^{2} d^{3} e^{4} - 9 \, a^{3} c d e^{6}\right )} e^{\left (-3\right )}}{c^{3} d^{3}}\right )} + \frac {{\left (5 \, c^{4} d^{8} - 12 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8}\right )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{128 \, \sqrt {c d} c^{2} d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________