Optimal. Leaf size=176 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} (1-k x)}{\sqrt [3]{1-k} \sqrt [3]{(1-x) x (1-k x)}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt [3]{1-k}}+\frac {\log (1-(2-k) x)}{2^{2/3} \sqrt [3]{1-k}}+\frac {\log (1-k x)}{2\ 2^{2/3} \sqrt [3]{1-k}}-\frac {3 \log \left (-1+k x+2^{2/3} \sqrt [3]{1-k} \sqrt [3]{(1-x) x (1-k x)}\right )}{2\ 2^{2/3} \sqrt [3]{1-k}} \]
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Rubi [F]
time = 0.28, 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 = {}
\begin {gather*} \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx &=\frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{1-k x}}{(1-x)^{2/3} x^{2/3} (1+(-2+k) x)} \, dx}{((1-x) x (1-k x))^{2/3}}\\ \end {align*}
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Mathematica [F]
time = 10.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {-k x +1}{\left (1+\left (-2+k \right ) x \right ) \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 932 vs.
\(2 (132) = 264\).
time = 36.70, size = 932, normalized size = 5.30 \begin {gather*} \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}}} \end {gather*}
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 \begin {gather*} - \int \frac {k x}{k x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} - 2 x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} + \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}}}\, dx - \int \left (- \frac {1}{k x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} - 2 x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} + \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}}}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\int \frac {k\,x-1}{\left (x\,\left (k-2\right )+1\right )\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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