Optimal. Leaf size=174 \[ \frac {1}{4} \sqrt [3]{6 x^2-2 x-1}-\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log \left (\sqrt [3]{6} 7^{2/3} \sqrt [3]{6 x^2-2 x-1}+7\right )+\frac {1}{24} \sqrt [3]{\frac {7}{6}} \log \left (6^{2/3} \sqrt [3]{7} \left (6 x^2-2 x-1\right )^{2/3}-\sqrt [3]{6} 7^{2/3} \sqrt [3]{6 x^2-2 x-1}+7\right )+\frac {\sqrt [3]{\frac {7}{2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {2}{7}} \sqrt [3]{6 x^2-2 x-1}}{\sqrt [6]{3}}\right )}{4\ 3^{5/6}} \]
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Rubi [A] time = 0.16, antiderivative size = 119, normalized size of antiderivative = 0.68, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {694, 266, 50, 58, 617, 204, 31} \begin {gather*} \frac {\sqrt [3]{(6 x-1)^2-7}}{4 \sqrt [3]{6}}+\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log (1-6 x)-\frac {1}{8} \sqrt [3]{\frac {7}{6}} \log \left (\sqrt [3]{(6 x-1)^2-7}+\sqrt [3]{7}\right )+\frac {\sqrt [3]{\frac {7}{2}} \tan ^{-1}\left (\frac {7-2\ 7^{2/3} \sqrt [3]{(6 x-1)^2-7}}{7 \sqrt {3}}\right )}{4\ 3^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 58
Rule 204
Rule 266
Rule 617
Rule 694
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {7}{6}+\frac {x^2}{6}}}{x} \, dx,x,-1+6 x\right )\\ &=\frac {1}{12} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {7}{6}+\frac {x}{6}}}{x} \, dx,x,(-1+6 x)^2\right )\\ &=\frac {\sqrt [3]{-7+(-1+6 x)^2}}{4 \sqrt [3]{6}}-\frac {7}{72} \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {7}{6}+\frac {x}{6}\right )^{2/3} x} \, dx,x,(-1+6 x)^2\right )\\ &=\frac {\sqrt [3]{-7+(-1+6 x)^2}}{4 \sqrt [3]{6}}+\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log (1-6 x)-\frac {1}{8} \sqrt [3]{\frac {7}{6}} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {7}{6}}+x} \, dx,x,\sqrt [3]{-1-2 x+6 x^2}\right )-\frac {1}{8} \left (\frac {7}{6}\right )^{2/3} \operatorname {Subst}\left (\int \frac {1}{\left (\frac {7}{6}\right )^{2/3}-\sqrt [3]{\frac {7}{6}} x+x^2} \, dx,x,\sqrt [3]{-1-2 x+6 x^2}\right )\\ &=\frac {\sqrt [3]{-7+(-1+6 x)^2}}{4 \sqrt [3]{6}}+\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log (1-6 x)-\frac {1}{8} \sqrt [3]{\frac {7}{6}} \log \left (\sqrt [3]{7}+\sqrt [3]{6} \sqrt [3]{-1-2 x+6 x^2}\right )-\frac {1}{4} \sqrt [3]{\frac {7}{6}} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{\frac {6}{7}} \sqrt [3]{-1-2 x+6 x^2}\right )\\ &=\frac {\sqrt [3]{-7+(-1+6 x)^2}}{4 \sqrt [3]{6}}+\frac {\sqrt [3]{\frac {7}{2}} \tan ^{-1}\left (\frac {7-2 \sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}}{7 \sqrt {3}}\right )}{4\ 3^{5/6}}+\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log (1-6 x)-\frac {1}{8} \sqrt [3]{\frac {7}{6}} \log \left (\sqrt [3]{7}+\sqrt [3]{6} \sqrt [3]{-1-2 x+6 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 152, normalized size = 0.87 \begin {gather*} \frac {1}{4} \sqrt [3]{6 x^2-2 x-1}+\frac {\sqrt [3]{\frac {7}{2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {2}{7}} \sqrt [3]{6 x^2-2 x-1}}{\sqrt [6]{3}}\right )}{4\ 3^{5/6}}-\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log \left (\sqrt [3]{(1-6 x)^2-7}+\sqrt [3]{7}\right )+\frac {1}{24} \sqrt [3]{\frac {7}{6}} \log \left (\left ((1-6 x)^2-7\right )^{2/3}-\sqrt [3]{7} \sqrt [3]{(1-6 x)^2-7}+7^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.43, size = 174, normalized size = 1.00 \begin {gather*} \frac {1}{4} \sqrt [3]{-1-2 x+6 x^2}+\frac {\sqrt [3]{\frac {7}{2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {2}{7}} \sqrt [3]{-1-2 x+6 x^2}}{\sqrt [6]{3}}\right )}{4\ 3^{5/6}}-\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log \left (7+\sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}\right )+\frac {1}{24} \sqrt [3]{\frac {7}{6}} \log \left (7-\sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}+6^{2/3} \sqrt [3]{7} \left (-1-2 x+6 x^2\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 146, normalized size = 0.84 \begin {gather*} \frac {1}{24} \cdot 6^{\frac {1}{6}} \sqrt {2} \left (-7\right )^{\frac {1}{3}} \arctan \left (\frac {1}{42} \cdot 6^{\frac {1}{6}} {\left (2 \cdot 6^{\frac {2}{3}} \sqrt {2} \left (-7\right )^{\frac {2}{3}} {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}} - 7 \cdot 6^{\frac {1}{3}} \sqrt {2}\right )}\right ) - \frac {1}{144} \cdot 6^{\frac {2}{3}} \left (-7\right )^{\frac {1}{3}} \log \left (6^{\frac {2}{3}} \left (-7\right )^{\frac {1}{3}} {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}} + 6^{\frac {1}{3}} \left (-7\right )^{\frac {2}{3}} + 6 \, {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{72} \cdot 6^{\frac {2}{3}} \left (-7\right )^{\frac {1}{3}} \log \left (-6^{\frac {2}{3}} \left (-7\right )^{\frac {1}{3}} + 6 \, {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{4} \, {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}} \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*} \int \frac {{\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}}}{6 \, x - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 16.08, size = 1308, normalized size = 7.52
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1308\) |
risch | \(\text {Expression too large to display}\) | \(2660\) |
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*} \int \frac {{\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}}}{6 \, x - 1}\,{d x} \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 {{\left (6\,x^2-2\,x-1\right )}^{1/3}}{6\,x-1} \,d x \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 {\sqrt [3]{6 x^{2} - 2 x - 1}}{6 x - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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