\[ \int \frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{16 c^{\frac {5}{2}} d^{\frac {5}{2}} e^{\frac {7}{2}}}+\frac {x^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{3 e}+\frac {\left (\left (-3 a \,e^{2}+5 c \,d^{2}\right ) \left (a \,e^{2}+3 c \,d^{2}\right )-2 c d e \left (-a \,e^{2}+5 c \,d^{2}\right ) x \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{24 c^{2} d^{2} e^{3}} \]
command
integrate(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{24} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, x e^{\left (-1\right )} - \frac {{\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} e^{\left (-3\right )}}{c^{2} d^{2}}\right )} x + \frac {{\left (15 \, c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} e^{\left (-3\right )}}{c^{2} d^{2}}\right )} + \frac {{\left (5 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{16 \, \sqrt {c d} c^{2} d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________