\[ \int \frac {x^6}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 x \left (d \,x^{3}+4 c \right )}{81 c \,d^{2} \left (-d \,x^{3}+8 c \right ) \sqrt {d \,x^{3}+c}}-\frac {2 \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c^{\frac {2}{3}}-c^{\frac {1}{3}} d^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}}{\left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{243 c \,d^{\frac {7}{3}} \sqrt {d \,x^{3}+c}\, \sqrt {\frac {c^{\frac {1}{3}} \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )}{\left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(x^6/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (d^{2} x^{6} - 7 \, c d x^{3} - 8 \, c^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (0, -\frac {4 \, c}{d}, x\right ) + {\left (d^{2} x^{4} + 4 \, c d x\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (c d^{5} x^{6} - 7 \, c^{2} d^{4} x^{3} - 8 \, c^{3} d^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d x^{3} + c} x^{6}}{d^{4} x^{12} - 14 \, c d^{3} x^{9} + 33 \, c^{2} d^{2} x^{6} + 112 \, c^{3} d x^{3} + 64 \, c^{4}}, x\right ) \]