\[ \int \frac {x}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )^{2}}{3 c^{\frac {1}{6}} \sqrt {d \,x^{3}+c}}\right )}{162 c^{\frac {11}{6}} d^{\frac {2}{3}}}-\frac {\arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{162 c^{\frac {11}{6}} d^{\frac {2}{3}}}-\frac {\arctan \left (\frac {c^{\frac {1}{6}} \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) \sqrt {3}}{\sqrt {d \,x^{3}+c}}\right ) \sqrt {3}}{162 c^{\frac {11}{6}} d^{\frac {2}{3}}}+\frac {2 x^{2}}{27 c^{2} \sqrt {d \,x^{3}+c}}-\frac {2 \sqrt {d \,x^{3}+c}}{27 c^{2} d^{\frac {2}{3}} \left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\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 ) \sqrt {2}\, \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}}}{81 c^{\frac {5}{3}} d^{\frac {2}{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}}}}+\frac {\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) \EllipticE \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 {1}{4}}}{27 c^{\frac {5}{3}} d^{\frac {2}{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/(-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 \[ {\rm integral}\left (-\frac {\sqrt {d x^{3} + c} x}{d^{3} x^{9} - 6 \, c d^{2} x^{6} - 15 \, c^{2} d x^{3} - 8 \, c^{3}}, x\right ) \]