\[ \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {d^{\frac {7}{3}} \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 )}{1024 c^{\frac {17}{6}}}-\frac {d^{\frac {7}{3}} \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{1024 c^{\frac {17}{6}}}-\frac {d^{\frac {7}{3}} \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}}{1024 c^{\frac {17}{6}}}-\frac {\sqrt {d \,x^{3}+c}}{56 c \,x^{7}}-\frac {19 d \sqrt {d \,x^{3}+c}}{1792 c^{2} x^{4}}+\frac {d^{2} \sqrt {d \,x^{3}+c}}{112 c^{3} x}-\frac {d^{\frac {7}{3}} \sqrt {d \,x^{3}+c}}{112 c^{3} \left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {d^{\frac {7}{3}} \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 {\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}} \sqrt {2}}{336 c^{\frac {8}{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 {3^{\frac {1}{4}} d^{\frac {7}{3}} \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}}}}{224 c^{\frac {8}{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((d*x^3+c)^(1/2)/x^8/(-d*x^3+8*c),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 \[ {\rm integral}\left (-\frac {\sqrt {d x^{3} + c}}{d x^{11} - 8 \, c x^{8}}, x\right ) \]