\[ \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (1-\left (2-k \right ) x \right ) 2^{\frac {1}{3}}}{2 \left (1-k \right )^{\frac {1}{3}}}+\frac {\ln \left (-k x +1\right ) 2^{\frac {1}{3}}}{4 \left (1-k \right )^{\frac {1}{3}}}-\frac {3 \ln \left (-1+k x +2^{\frac {2}{3}} \left (1-k \right )^{\frac {1}{3}} \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{4 \left (1-k \right )^{\frac {1}{3}}}-\frac {\arctan \left (\frac {\left (1+\frac {2^{\frac {1}{3}} \left (-k x +1\right )}{\left (1-k \right )^{\frac {1}{3}} \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}{2 \left (1-k \right )^{\frac {1}{3}}} \]
command
integrate((-k*x+1)/(1+(-2+k)*x)/((1-x)*x*(-k*x+1))^(2/3),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\frac {24 \, \sqrt {3} 2^{\frac {1}{3}} {\left ({\left (k^{5} - 3 \, k^{4} - 4 \, k^{3} + 22 \, k^{2} - 24 \, k + 8\right )} x^{4} - 2 \, {\left (k^{4} - 10 \, k^{3} + 27 \, k^{2} - 26 \, k + 8\right )} x^{3} - 6 \, {\left (k^{3} - 4 \, k^{2} + 4 \, k - 1\right )} x^{2} - 2 \, {\left (k^{2} - 1\right )} x + k - 1\right )} {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {2}{3}}}{{\left (k - 1\right )}^{\frac {1}{3}}} - \frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left ({\left (k^{6} + 27 \, k^{5} - 40 \, k^{4} - 20 \, k^{3} + 48 \, k^{2} - 16 \, k\right )} x^{5} - {\left (33 \, k^{5} + 55 \, k^{4} - 220 \, k^{3} + 132 \, k^{2} + 16 \, k - 16\right )} x^{4} + 2 \, {\left (55 \, k^{4} - 55 \, k^{3} - 66 \, k^{2} + 82 \, k - 16\right )} x^{3} - 2 \, {\left (55 \, k^{3} - 99 \, k^{2} + 38 \, k + 6\right )} x^{2} + {\left (33 \, k^{2} - 61 \, k + 28\right )} x - k + 1\right )} {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {1}{3}}}{{\left (k - 1\right )}^{\frac {2}{3}}} + \sqrt {3} {\left ({\left (k^{6} - 48 \, k^{5} - 192 \, k^{4} + 416 \, k^{3} - 48 \, k^{2} - 192 \, k + 64\right )} x^{6} + 6 \, {\left (7 \, k^{5} + 104 \, k^{4} - 80 \, k^{3} - 176 \, k^{2} + 176 \, k - 32\right )} x^{5} - 3 \, {\left (139 \, k^{4} + 256 \, k^{3} - 768 \, k^{2} + 352 \, k + 16\right )} x^{4} + 4 \, {\left (203 \, k^{3} - 192 \, k^{2} - 120 \, k + 104\right )} x^{3} - 3 \, {\left (139 \, k^{2} - 208 \, k + 64\right )} x^{2} + 6 \, {\left (7 \, k - 8\right )} x + 1\right )}}{3 \, {\left ({\left (k^{6} + 96 \, k^{5} - 48 \, k^{4} - 160 \, k^{3} + 240 \, k^{2} - 192 \, k + 64\right )} x^{6} - 6 \, {\left (17 \, k^{5} + 64 \, k^{4} - 112 \, k^{3} + 80 \, k^{2} - 80 \, k + 32\right )} x^{5} + 3 \, {\left (149 \, k^{4} + 32 \, k^{3} - 96 \, k^{2} - 160 \, k + 80\right )} x^{4} - 4 \, {\left (157 \, k^{3} - 24 \, k^{2} - 168 \, k + 40\right )} x^{3} + 3 \, {\left (149 \, k^{2} - 128 \, k - 16\right )} x^{2} - 6 \, {\left (17 \, k - 16\right )} x + 1\right )}}\right )}{6 \, {\left (k - 1\right )}^{\frac {1}{3}}} - \frac {2^{\frac {1}{3}} \log \left (\frac {\frac {12 \cdot 2^{\frac {2}{3}} {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {2}{3}} {\left ({\left (k^{3} + k^{2} - 4 \, k + 2\right )} x^{2} - 2 \, {\left (2 \, k^{2} - 3 \, k + 1\right )} x + k - 1\right )}}{{\left (k - 1\right )}^{\frac {2}{3}}} + 6 \, {\left ({\left (k^{3} + 8 \, k^{2} - 8 \, k\right )} x^{3} - {\left (11 \, k^{2} - 8\right )} x^{2} + {\left (11 \, k - 8\right )} x - 1\right )} {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {1}{3}} + \frac {2^{\frac {1}{3}} {\left ({\left (k^{4} + 28 \, k^{3} - 12 \, k^{2} - 32 \, k + 16\right )} x^{4} - 4 \, {\left (8 \, k^{3} + 15 \, k^{2} - 30 \, k + 8\right )} x^{3} + 6 \, {\left (13 \, k^{2} - 10 \, k - 2\right )} x^{2} - 4 \, {\left (8 \, k - 7\right )} x + 1\right )}}{{\left (k - 1\right )}^{\frac {1}{3}}}}{{\left (k^{4} - 8 \, k^{3} + 24 \, k^{2} - 32 \, k + 16\right )} x^{4} + 4 \, {\left (k^{3} - 6 \, k^{2} + 12 \, k - 8\right )} x^{3} + 6 \, {\left (k^{2} - 4 \, k + 4\right )} x^{2} + 4 \, {\left (k - 2\right )} x + 1}\right )}{12 \, {\left (k - 1\right )}^{\frac {1}{3}}} + \frac {2^{\frac {1}{3}} \log \left (\frac {\frac {6 \cdot 2^{\frac {1}{3}} {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {1}{3}} {\left (k x - 1\right )}}{{\left (k - 1\right )}^{\frac {1}{3}}} - \frac {2^{\frac {2}{3}} {\left ({\left (k^{2} - 4 \, k + 4\right )} x^{2} + 2 \, {\left (k - 2\right )} x + 1\right )}}{{\left (k - 1\right )}^{\frac {2}{3}}} - 12 \, {\left (k x^{3} - {\left (k + 1\right )} x^{2} + x\right )}^{\frac {2}{3}}}{{\left (k^{2} - 4 \, k + 4\right )} x^{2} + 2 \, {\left (k - 2\right )} x + 1}\right )}{6 \, {\left (k - 1\right )}^{\frac {1}{3}}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]