\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{3 e}+\frac {\left (-a \,e^{2}+c \,d^{2}\right )^{3} \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{16 c^{\frac {3}{2}} d^{\frac {3}{2}} e^{\frac {5}{2}}}+\frac {\left (\frac {a}{c d}-\frac {d}{e^{2}}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{8} \]
command
integrate((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
\[ \frac {1}{24} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, c d x + \frac {{\left (c^{3} d^{4} e + 7 \, a c^{2} d^{2} e^{3}\right )} e^{\left (-2\right )}}{c^{2} d^{2}}\right )} x - \frac {{\left (3 \, c^{3} d^{5} - 8 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}\right )} e^{\left (-2\right )}}{c^{2} d^{2}}\right )} - \frac {{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{16 \, \sqrt {c d} c d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________