\[ \int \frac {x^3}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {x^{4} F_{1}\left (\frac {4}{3}, \frac {3}{2}, 1, \frac {7}{3}, -\frac {d \,x^{3}}{c}, \frac {d \,x^{3}}{8 c}\right ) \sqrt {1+\frac {d \,x^{3}}{c}}}{32 c^{2} \sqrt {d \,x^{3}+c}} \]
command
integrate(x^3/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (-\frac {\sqrt {d x^{3} + c} x^{3}}{d^{3} x^{9} - 6 \, c d^{2} x^{6} - 15 \, c^{2} d x^{3} - 8 \, c^{3}}, x\right ) \]