\[ \int \frac {x^3}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx \]
Optimal antiderivative \[ \frac {3 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{8 c^{\frac {5}{2}} d^{\frac {5}{2}} e^{\frac {7}{2}}}-\frac {3 \left (a \,e^{2}+3 c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c^{2} d^{2} e^{3}}-\frac {2 d^{3} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{e^{3} \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )}+\frac {\left (e x +d \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{2 c d \,e^{3}} \]
command
integrate(x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, d^{3} e^{\left (-3\right )}}{\sqrt {c d} d e^{\frac {1}{2}} + {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} e} + \frac {1}{4} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (\frac {2 \, x e^{\left (-2\right )}}{c d} - \frac {{\left (7 \, c d^{2} e^{5} + 3 \, a e^{7}\right )} e^{\left (-8\right )}}{c^{2} d^{2}}\right )} - \frac {3 \, {\left (5 \, \sqrt {c d} c^{2} d^{4} e^{\frac {1}{2}} + 2 \, \sqrt {c d} a c d^{2} e^{\frac {5}{2}} + \sqrt {c d} a^{2} e^{\frac {9}{2}}\right )} e^{\left (-4\right )} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right )}{8 \, c^{3} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________