\[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {3}\, \arctan \left (\frac {\frac {\sqrt {3}}{3}-\frac {2 x \sqrt {3}}{3}+\frac {\sqrt {3}\, x^{2}}{3}+\frac {2 \left (x +\left (-1-k \right ) x^{2}+k \,x^{3}\right )^{\frac {2}{3}} \sqrt {3}}{3 b^{\frac {1}{3}}}}{\left (-1+x \right )^{2}}\right )}{2 b^{\frac {2}{3}}}-\frac {\ln \left (b^{\frac {1}{6}}-b^{\frac {1}{6}} x +\left (x +\left (-1-k \right ) x^{2}+k \,x^{3}\right )^{\frac {1}{3}}\right )}{2 b^{\frac {2}{3}}}-\frac {\ln \left (-b^{\frac {1}{6}}+b^{\frac {1}{6}} x +\left (x +\left (-1-k \right ) x^{2}+k \,x^{3}\right )^{\frac {1}{3}}\right )}{2 b^{\frac {2}{3}}}+\frac {\ln \left (b^{\frac {1}{3}}-2 b^{\frac {1}{3}} x +b^{\frac {1}{3}} x^{2}+\left (b^{\frac {1}{6}}-b^{\frac {1}{6}} x \right ) \left (x +\left (-1-k \right ) x^{2}+k \,x^{3}\right )^{\frac {1}{3}}+\left (x +\left (-1-k \right ) x^{2}+k \,x^{3}\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}}}+\frac {\ln \left (b^{\frac {1}{3}}-2 b^{\frac {1}{3}} x +b^{\frac {1}{3}} x^{2}+\left (-b^{\frac {1}{6}}+b^{\frac {1}{6}} x \right ) \left (x +\left (-1-k \right ) x^{2}+k \,x^{3}\right )^{\frac {1}{3}}+\left (x +\left (-1-k \right ) x^{2}+k \,x^{3}\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}}} \]
command
Integrate[((-1 + (-1 + 2*k)*x)*(1 - 2*x + x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + 4*b*x + (1 - 6*b)*x^2 + (4*b - 2*k)*x^3 + (-b + k^2)*x^4)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{k-\frac {1}{x}} (-1+x) \left (-2 \sqrt {3} \text {ArcTan}\left (\frac {\left (k-\frac {1}{x}\right )^{2/3}+2 \sqrt [3]{b} \left (-1+\frac {1}{x}\right )^{4/3}}{\sqrt {3} \left (k-\frac {1}{x}\right )^{2/3}}\right )-2 \log \left (\sqrt [3]{k-\frac {1}{x}}-\sqrt [6]{b} \left (-1+\frac {1}{x}\right )^{2/3}\right )-2 \log \left (\sqrt [3]{k-\frac {1}{x}}+\sqrt [6]{b} \left (-1+\frac {1}{x}\right )^{2/3}\right )+\log \left (\left (k-\frac {1}{x}\right )^{2/3}-\sqrt [6]{b} \sqrt [3]{k-\frac {1}{x}} \left (-1+\frac {1}{x}\right )^{2/3}+\sqrt [3]{b} \left (-1+\frac {1}{x}\right )^{4/3}\right )+\log \left (\left (k-\frac {1}{x}\right )^{2/3}+\sqrt [6]{b} \sqrt [3]{k-\frac {1}{x}} \left (-1+\frac {1}{x}\right )^{2/3}+\sqrt [3]{b} \left (-1+\frac {1}{x}\right )^{4/3}\right )\right )}{4 b^{2/3} \left (-1+\frac {1}{x}\right )^{2/3} \sqrt [3]{(-1+x) x (-1+k x)}} \]
Mathematica 12.3 output
\[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx \]________________________________________________________________________________________