Integrand size = 22, antiderivative size = 174 \[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\frac {1}{4} \sqrt [3]{-1-2 x+6 x^2}+\frac {\sqrt [3]{\frac {7}{2}} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {2}{7}} \sqrt [3]{-1-2 x+6 x^2}}{\sqrt [6]{3}}\right )}{4\ 3^{5/6}}-\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log \left (7+\sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}\right )+\frac {1}{24} \sqrt [3]{\frac {7}{6}} \log \left (7-\sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}+6^{2/3} \sqrt [3]{7} \left (-1-2 x+6 x^2\right )^{2/3}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.68, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {708, 272, 52, 60, 631, 210, 31} \[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\frac {\sqrt [3]{\frac {7}{2}} \arctan \left (\frac {7-2\ 7^{2/3} \sqrt [3]{(6 x-1)^2-7}}{7 \sqrt {3}}\right )}{4\ 3^{5/6}}+\frac {\sqrt [3]{(6 x-1)^2-7}}{4 \sqrt [3]{6}}+\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log (1-6 x)-\frac {1}{8} \sqrt [3]{\frac {7}{6}} \log \left (\sqrt [3]{(6 x-1)^2-7}+\sqrt [3]{7}\right ) \]
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Rule 31
Rule 52
Rule 60
Rule 210
Rule 272
Rule 631
Rule 708
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {\sqrt [3]{-\frac {7}{6}+\frac {x^2}{6}}}{x} \, dx,x,-1+6 x\right ) \\ & = \frac {1}{12} \text {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} \text {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}} \text {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} \text {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}} \text {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}} \arctan \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*}
Time = 0.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\frac {1}{4} \sqrt [3]{-1-2 x+6 x^2}+\frac {\sqrt [3]{\frac {7}{2}} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {2}{7}} \sqrt [3]{-1-2 x+6 x^2}}{\sqrt [6]{3}}\right )}{4\ 3^{5/6}}-\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log \left (7+\sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}\right )+\frac {1}{24} \sqrt [3]{\frac {7}{6}} \log \left (7-\sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}+6^{2/3} \sqrt [3]{7} \left (-1-2 x+6 x^2\right )^{2/3}\right ) \]
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Time = 14.81 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {\left (6 x^{2}-2 x -1\right )^{\frac {1}{3}}}{4}+\frac {7^{\frac {1}{3}} 6^{\frac {2}{3}} \ln \left (6\right )}{144}-\frac {7^{\frac {1}{3}} 6^{\frac {2}{3}} \ln \left (7^{\frac {1}{3}} 6^{\frac {2}{3}}+6 \left (6 x^{2}-2 x -1\right )^{\frac {1}{3}}\right )}{72}+\frac {7^{\frac {1}{3}} 6^{\frac {2}{3}} \ln \left (-7^{\frac {1}{3}} 6^{\frac {2}{3}} \left (6 x^{2}-2 x -1\right )^{\frac {1}{3}}+7^{\frac {2}{3}} 6^{\frac {1}{3}}+6 \left (6 x^{2}-2 x -1\right )^{\frac {2}{3}}\right )}{144}-\frac {7^{\frac {1}{3}} 3^{\frac {1}{6}} 2^{\frac {2}{3}} \arctan \left (\frac {2 \sqrt {3}\, \left (6 x^{2}-2 x -1\right )^{\frac {1}{3}} 7^{\frac {2}{3}} 6^{\frac {1}{3}}}{21}-\frac {\sqrt {3}}{3}\right )}{24}\) | \(149\) |
trager | \(\text {Expression too large to display}\) | \(1309\) |
risch | \(\text {Expression too large to display}\) | \(2660\) |
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Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\frac {1}{24} \cdot 6^{\frac {1}{6}} \sqrt {2} \left (-7\right )^{\frac {1}{3}} \arctan \left (\frac {1}{42} \cdot 6^{\frac {1}{6}} {\left (2 \cdot 6^{\frac {2}{3}} \sqrt {2} \left (-7\right )^{\frac {2}{3}} {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}} - 7 \cdot 6^{\frac {1}{3}} \sqrt {2}\right )}\right ) - \frac {1}{144} \cdot 6^{\frac {2}{3}} \left (-7\right )^{\frac {1}{3}} \log \left (6^{\frac {2}{3}} \left (-7\right )^{\frac {1}{3}} {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}} + 6^{\frac {1}{3}} \left (-7\right )^{\frac {2}{3}} + 6 \, {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{72} \cdot 6^{\frac {2}{3}} \left (-7\right )^{\frac {1}{3}} \log \left (-6^{\frac {2}{3}} \left (-7\right )^{\frac {1}{3}} + 6 \, {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{4} \, {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}} \]
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\[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\int \frac {\sqrt [3]{6 x^{2} - 2 x - 1}}{6 x - 1}\, dx \]
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\[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\int { \frac {{\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}}}{6 \, x - 1} \,d x } \]
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\[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\int { \frac {{\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}}}{6 \, x - 1} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\int \frac {{\left (6\,x^2-2\,x-1\right )}^{1/3}}{6\,x-1} \,d x \]
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