\[ \int \frac {1}{x^2 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {d^{\frac {1}{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 )}{3888 c^{\frac {23}{6}}}-\frac {d^{\frac {1}{3}} \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3888 c^{\frac {23}{6}}}-\frac {d^{\frac {1}{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}}{3888 c^{\frac {23}{6}}}+\frac {5}{648 c^{3} x \sqrt {d \,x^{3}+c}}+\frac {1}{216 c^{2} x \left (-d \,x^{3}+8 c \right ) \sqrt {d \,x^{3}+c}}-\frac {31 \sqrt {d \,x^{3}+c}}{1296 c^{4} x}+\frac {31 d^{\frac {1}{3}} \sqrt {d \,x^{3}+c}}{1296 c^{4} \left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {31 d^{\frac {1}{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}}{3888 c^{\frac {11}{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 {31 d^{\frac {1}{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}}}\, 3^{\frac {1}{4}}}{2592 c^{\frac {11}{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(1/x^2/(-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
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {\sqrt {d x^{3} + c}}{d^{4} x^{14} - 14 \, c d^{3} x^{11} + 33 \, c^{2} d^{2} x^{8} + 112 \, c^{3} d x^{5} + 64 \, c^{4} x^{2}}, x\right ) \]