\[ \int \frac {c+d x}{\sqrt {-1+x^3}} \, dx \]
Optimal antiderivative \[ -\frac {2 d \sqrt {x^{3}-1}}{1-x -\sqrt {3}}-\frac {2 \left (1-x \right ) \EllipticF \left (\frac {1-x +\sqrt {3}}{1-x -\sqrt {3}}, 2 i-i \sqrt {3}\right ) \left (c +d +d \sqrt {3}\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}+x +1}{\left (1-x -\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{3 \sqrt {x^{3}-1}\, \sqrt {\frac {-1+x}{\left (1-x -\sqrt {3}\right )^{2}}}}+\frac {3^{\frac {1}{4}} d \left (1-x \right ) \EllipticE \left (\frac {1-x +\sqrt {3}}{1-x -\sqrt {3}}, 2 i-i \sqrt {3}\right ) \sqrt {\frac {x^{2}+x +1}{\left (1-x -\sqrt {3}\right )^{2}}}\, \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-1}\, \sqrt {\frac {-1+x}{\left (1-x -\sqrt {3}\right )^{2}}}} \]
command
integrate((d*x+c)/(x^3-1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ 2 \, c {\rm weierstrassPInverse}\left (0, 4, x\right ) - 2 \, d {\rm weierstrassZeta}\left (0, 4, {\rm weierstrassPInverse}\left (0, 4, x\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {d x + c}{\sqrt {x^{3} - 1}}, x\right ) \]