\[ \int \frac {1}{x^3 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx \]
Optimal antiderivative \[ -\frac {F_{1}\left (-\frac {2}{3}, \frac {1}{2}, 2, \frac {1}{3}, -\frac {d \,x^{3}}{c}, \frac {d \,x^{3}}{8 c}\right ) \sqrt {1+\frac {d \,x^{3}}{c}}}{128 c^{2} x^{2} \sqrt {d \,x^{3}+c}} \]
command
integrate(1/x^3/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),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 \[ {\rm integral}\left (\frac {\sqrt {d x^{3} + c}}{d^{3} x^{12} - 15 \, c d^{2} x^{9} + 48 \, c^{2} d x^{6} + 64 \, c^{3} x^{3}}, x\right ) \]