\[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (5 a^{3} e^{6}+9 a^{2} c \,d^{2} e^{4}+15 a \,c^{2} d^{4} e^{2}+35 c^{3} d^{6}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{128 c^{\frac {7}{2}} d^{\frac {7}{2}} e^{\frac {9}{2}}}+\frac {\left (\frac {a}{c d}-\frac {7 d}{e^{2}}\right ) x^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{24}+\frac {x^{3} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 e}-\frac {\left (105 c^{3} d^{6}-25 a \,c^{2} d^{4} e^{2}-17 a^{2} c \,d^{2} e^{4}-15 a^{3} e^{6}-2 c d e \left (-5 a^{2} e^{4}-6 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) x \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{192 c^{3} d^{3} e^{4}} \]
command
integrate(x^3*(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
\[ \frac {1}{192} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (6 \, x e^{\left (-1\right )} - \frac {{\left (7 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} e^{\left (-4\right )}}{c^{3} d^{3}}\right )} x + \frac {{\left (35 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 5 \, a^{2} c d e^{5}\right )} e^{\left (-4\right )}}{c^{3} d^{3}}\right )} x - \frac {{\left (105 \, c^{3} d^{6} - 25 \, a c^{2} d^{4} e^{2} - 17 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6}\right )} e^{\left (-4\right )}}{c^{3} d^{3}}\right )} - \frac {{\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} e^{\left (-\frac {9}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{128 \, \sqrt {c d} c^{3} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________