Optimal. Leaf size=111 \[ \frac {\log (x)}{2 \sqrt [3]{k}}+\frac {\log (1-(k+1) x)}{2 \sqrt [3]{k}}-\frac {3 \log \left (\sqrt [3]{(1-x) x (1-k x)}-\sqrt [3]{k} x\right )}{2 \sqrt [3]{k}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{k} x}{\sqrt [3]{(1-x) x (1-k x)}}+1}{\sqrt {3}}\right )}{\sqrt [3]{k}} \]
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Rubi [F] time = 0.61, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {2-(1+k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(1+k) x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {2-(1+k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(1+k) x)} \, dx &=\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {2-(1+k) x}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} (1-(1+k) x)} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}\\ &=\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{x} (1+(-1-k) x) \sqrt [3]{1-k x}} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}\\ &=\frac {3 \sqrt [3]{1-x} x \sqrt [3]{1-k x} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};x,k x\right )}{2 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{x} (1+(-1-k) x) \sqrt [3]{1-k x}} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}\\ \end {align*}
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Mathematica [F] time = 1.64, size = 0, normalized size = 0.00 \[ \int \frac {2-(1+k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(1+k) x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (k + 1\right )} x - 2}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left ({\left (k + 1\right )} x - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {-\left (k +1\right ) x +2}{\left (\left (-x +1\right ) \left (-k x +1\right ) x \right )^{\frac {1}{3}} \left (-\left (k +1\right ) x +1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (k + 1\right )} x - 2}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left ({\left (k + 1\right )} x - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (k+1\right )-2}{\left (x\,\left (k+1\right )-1\right )\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {k x + x - 2}{\sqrt [3]{x \left (x - 1\right ) \left (k x - 1\right )} \left (k x + x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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