\[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\sqrt {-a-b x^3}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (\frac {b}{a}\right )^{\frac {2}{3}} \sqrt {-b \,x^{3}-a}}{b \left (1+\left (\frac {b}{a}\right )^{\frac {1}{3}} x -\sqrt {3}\right )}+\frac {3^{\frac {1}{4}} \left (1+\left (\frac {b}{a}\right )^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {1+\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\sqrt {3}}{1+\left (\frac {b}{a}\right )^{\frac {1}{3}} x -\sqrt {3}}, 2 i-i \sqrt {3}\right ) \sqrt {\frac {1-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}} x^{2}}{\left (1+\left (\frac {b}{a}\right )^{\frac {1}{3}} x -\sqrt {3}\right )^{2}}}\, \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\left (\frac {b}{a}\right )^{\frac {1}{3}} \sqrt {-b \,x^{3}-a}\, \sqrt {\frac {-1-\left (\frac {b}{a}\right )^{\frac {1}{3}} x}{\left (1+\left (\frac {b}{a}\right )^{\frac {1}{3}} x -\sqrt {3}\right )^{2}}}} \]
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
\[ {\rm integral}\left (-\frac {\sqrt {-b x^{3} - a} x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \sqrt {-b x^{3} - a} {\left (\sqrt {3} + 1\right )}}{b x^{3} + a}, x\right ) \]