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