\[ \int \frac {x^4 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx \]
Optimal antiderivative \[ \frac {x^{2} \left (d \,x^{3}+c \right )^{\frac {3}{2}}}{3 d \left (-d \,x^{3}+8 c \right )}-\frac {9 c^{\frac {7}{6}} \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 )}{d^{\frac {5}{3}}}+\frac {9 c^{\frac {7}{6}} \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{d^{\frac {5}{3}}}+\frac {9 c^{\frac {7}{6}} \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}}{d^{\frac {5}{3}}}+\frac {13 x^{2} \sqrt {d \,x^{3}+c}}{21 d}+\frac {265 c \sqrt {d \,x^{3}+c}}{7 d^{\frac {5}{3}} \left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {265 c^{\frac {4}{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 {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}}}{21 d^{\frac {5}{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 {265 \,3^{\frac {1}{4}} c^{\frac {4}{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}}}}{14 d^{\frac {5}{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^4*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^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 \[ {\rm integral}\left (\frac {{\left (d x^{7} + c x^{4}\right )} \sqrt {d x^{3} + c}}{d^{2} x^{6} - 16 \, c d x^{3} + 64 \, c^{2}}, x\right ) \]