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