\[ \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt {-a+b x^3}} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {b \,x^{3}-a}}{b^{\frac {1}{3}} \left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right )}-\frac {3^{\frac {1}{4}} a^{\frac {1}{3}} \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}{-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}, 2 i-i \sqrt {3}\right ) \sqrt {\frac {a^{\frac {2}{3}}+a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right )^{2}}}\, \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{b^{\frac {1}{3}} \sqrt {b \,x^{3}-a}\, \sqrt {-\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right )}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((-b^(1/3)*x+a^(1/3)*(1+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
\[ \frac {2 \, {\left (a^{\frac {1}{3}} \sqrt {b} {\left (\sqrt {3} + 1\right )} {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right ) + b^{\frac {5}{6}} {\rm weierstrassZeta}\left (0, \frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right )\right )\right )}}{b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {b^{\frac {1}{3}} x - a^{\frac {1}{3}} {\left (\sqrt {3} + 1\right )}}{\sqrt {b x^{3} - a}}, x\right ) \]