Optimal. Leaf size=161 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{k} x}{\sqrt [3]{k} x+2 \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{k}}-\frac {\log \left (-\sqrt [3]{k} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{k}}+\frac {\log \left (k^{2/3} x^2+\sqrt [3]{k} x \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{k}} \]
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Rubi [F]
time = 0.42, 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 {2-(1+k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(1+k) x)} \, dx \end {gather*}
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 = 41.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2-(1+k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(1+k) x)} \, 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 \[\int \frac {2-\left (1+k \right ) x}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (1-\left (1+k \right ) x \right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 + x - 2}{\sqrt [3]{x \left (x - 1\right ) \left (k x - 1\right )} \left (k x + x - 1\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 {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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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