\[ \int \frac {x}{(c+d x) \sqrt {c^3-8 d^3 x^3}} \, dx \]
Optimal antiderivative \[ \frac {2 \arctanh \left (\frac {\left (-2 d x +c \right )^{2}}{3 \sqrt {c}\, \sqrt {-8 d^{3} x^{3}+c^{3}}}\right )}{9 d^{2} \sqrt {c}}-\frac {\left (-2 d x +c \right ) \EllipticF \left (\frac {-2 d x +c \left (1-\sqrt {3}\right )}{-2 d x +c \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {4 d^{2} x^{2}+2 c d x +c^{2}}{\left (-2 d x +c \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{9 d^{2} \sqrt {-8 d^{3} x^{3}+c^{3}}\, \sqrt {\frac {c \left (-2 d x +c \right )}{\left (-2 d x +c \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(x/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {\sqrt {c} d^{2} \log \left (\frac {8 \, d^{6} x^{6} - 240 \, c d^{5} x^{5} + 408 \, c^{2} d^{4} x^{4} + 88 \, c^{3} d^{3} x^{3} + 156 \, c^{4} d^{2} x^{2} + 12 \, c^{5} d x + 17 \, c^{6} + 3 \, {\left (8 \, d^{4} x^{4} - 52 \, c d^{3} x^{3} + 12 \, c^{2} d^{2} x^{2} - 4 \, c^{3} d x + 5 \, c^{4}\right )} \sqrt {-8 \, d^{3} x^{3} + c^{3}} \sqrt {c}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right ) - 3 \, \sqrt {2} \sqrt {-d^{3}} c {\rm weierstrassPInverse}\left (0, \frac {c^{3}}{2 \, d^{3}}, x\right )}{18 \, c d^{4}}, \frac {2 \, \sqrt {-c} d^{2} \arctan \left (\frac {{\left (4 \, d^{3} x^{3} - 24 \, c d^{2} x^{2} - 6 \, c^{2} d x - 5 \, c^{3}\right )} \sqrt {-8 \, d^{3} x^{3} + c^{3}} \sqrt {-c}}{3 \, {\left (16 \, c d^{4} x^{4} - 8 \, c^{2} d^{3} x^{3} - 2 \, c^{4} d x + c^{5}\right )}}\right ) - 3 \, \sqrt {2} \sqrt {-d^{3}} c {\rm weierstrassPInverse}\left (0, \frac {c^{3}}{2 \, d^{3}}, x\right )}{18 \, c d^{4}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-8 \, d^{3} x^{3} + c^{3}} x}{8 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} - c^{3} d x - c^{4}}, x\right ) \]