\[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx \]
Optimal antiderivative \[ \frac {\left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}{-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}, 2 i-i \sqrt {3}\right ) \left (b^{\frac {1}{3}} e +a^{\frac {1}{3}} f \left (1+\sqrt {3}\right )\right ) \sqrt {\frac {a^{\frac {2}{3}}+a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right )^{2}}}\, \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) 3^{\frac {1}{4}}}{3 a^{\frac {1}{3}} b^{\frac {2}{3}} \sqrt {b \,x^{3}-a}\, \sqrt {-\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right )}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right )^{2}}}}+\frac {\arctan \left (\frac {a^{\frac {1}{6}} \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \sqrt {-3+2 \sqrt {3}}}{\sqrt {b \,x^{3}-a}}\right ) \left (b^{\frac {1}{3}} e +a^{\frac {1}{3}} f \left (1-\sqrt {3}\right )\right )}{b^{\frac {2}{3}} \sqrt {a}\, \sqrt {-9+6 \sqrt {3}}} \]
command
integrate((f*x+e)/(-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
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (-\frac {\sqrt {b x^{3} - a} {\left (2 \, {\left (2 \, b f x^{4} + 2 \, b e x^{3} - 2 \, a f x - 2 \, a e - \sqrt {3} {\left (b f x^{4} + b e x^{3} + 2 \, a f x + 2 \, a e\right )}\right )} a^{\frac {2}{3}} + {\left (b f x^{5} + b e x^{4} + 8 \, a f x^{2} + 8 \, a e x - \sqrt {3} {\left (b f x^{5} + b e x^{4} - 4 \, a f x^{2} - 4 \, a e x\right )}\right )} a^{\frac {1}{3}} b^{\frac {1}{3}} + {\left (b f x^{6} + b e x^{5} - 10 \, a f x^{3} - 10 \, a e x^{2} - 6 \, \sqrt {3} {\left (a f x^{3} + a e x^{2}\right )}\right )} b^{\frac {2}{3}}\right )}}{b^{3} x^{9} - 21 \, a b^{2} x^{6} + 12 \, a^{2} b x^{3} + 8 \, a^{3}}, x\right ) \]