Integrand size = 21, antiderivative size = 201 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=6 \sqrt [6]{x}+x-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {1-\sqrt {5}+4 \sqrt [6]{x}}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}+4 \sqrt [6]{x}\right )\right )+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1598, 348, 308, 208, 648, 632, 210, 642, 31} \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 \sqrt [6]{x}-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (4 \sqrt [6]{x}+\sqrt {5}+1\right )\right )+x+6 \sqrt [6]{x}+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt {5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt {5} \sqrt [6]{x}+\sqrt [6]{x}+2\right ) \]
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Rule 31
Rule 208
Rule 210
Rule 308
Rule 348
Rule 632
Rule 642
Rule 648
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{5/6}}{-1+x^{5/6}} \, dx \\ & = 6 \text {Subst}\left (\int \frac {x^{10}}{-1+x^5} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \text {Subst}\left (\int \left (1+x^5+\frac {1}{-1+x^5}\right ) \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}+x+6 \text {Subst}\left (\int \frac {1}{-1+x^5} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}+x-\frac {6}{5} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [6]{x}\right )-\frac {12}{5} \text {Subst}\left (\int \frac {1+\frac {1}{4} \left (1-\sqrt {5}\right ) x}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {12}{5} \text {Subst}\left (\int \frac {1+\frac {1}{4} \left (1+\sqrt {5}\right ) x}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}+x+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {1}{10} \left (3 \left (1-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (1-\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {1}{10} \left (3 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {1}{10} \left (3 \left (1+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (1+\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {1}{10} \left (3 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}+x+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )+\frac {1}{5} \left (3 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5+\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (1+\sqrt {5}\right )+2 \sqrt [6]{x}\right )+\frac {1}{5} \left (3 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5-\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (1-\sqrt {5}\right )+2 \sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}+x-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {1-\sqrt {5}+4 \sqrt [6]{x}}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}+4 \sqrt [6]{x}\right )\right )+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=6 \sqrt [6]{x}+x+\frac {6}{5} \log \left (-1+\sqrt [6]{x}\right )-\frac {6}{5} \text {RootSum}\left [1+\text {$\#$1}+\text {$\#$1}^2+\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {4 \log \left (\sqrt [6]{x}-\text {$\#$1}\right )+3 \log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{1+2 \text {$\#$1}+3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
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Time = 1.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.66
method | result | size |
meijerg | \(\frac {6 \left (-1\right )^{\frac {4}{5}} \left (-\frac {5 x^{\frac {1}{6}} \left (-1\right )^{\frac {1}{5}} \left (11 x^{\frac {5}{6}}+66\right )}{66}-\left (-1\right )^{\frac {1}{5}} \left (\ln \left (1-x^{\frac {1}{6}}\right )+\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{5}\right ) x^{\frac {1}{6}}+x^{\frac {1}{3}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) x^{\frac {1}{6}}}{1-\cos \left (\frac {2 \pi }{5}\right ) x^{\frac {1}{6}}}\right )-\cos \left (\frac {\pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{5}\right ) x^{\frac {1}{6}}+x^{\frac {1}{3}}\right )-2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) x^{\frac {1}{6}}}{1+\cos \left (\frac {\pi }{5}\right ) x^{\frac {1}{6}}}\right )\right )\right )}{5}\) | \(132\) |
derivativedivides | \(x +6 x^{\frac {1}{6}}+\frac {6 \ln \left (x^{\frac {1}{6}}-1\right )}{5}-\frac {3 \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {5}\right ) \left (-\sqrt {5}+1\right )}{10}-\frac {12 \left (4-\frac {\left (-\sqrt {5}+1\right )^{2}}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {3 \left (-\sqrt {5}-1\right ) \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {5}\right )}{10}+\frac {12 \left (-4-\frac {\left (-\sqrt {5}-1\right ) \left (\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}+\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}\) | \(166\) |
default | \(x +6 x^{\frac {1}{6}}+\frac {6 \ln \left (x^{\frac {1}{6}}-1\right )}{5}-\frac {3 \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {5}\right ) \left (-\sqrt {5}+1\right )}{10}-\frac {12 \left (4-\frac {\left (-\sqrt {5}+1\right )^{2}}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {3 \left (-\sqrt {5}-1\right ) \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {5}\right )}{10}+\frac {12 \left (-4-\frac {\left (-\sqrt {5}-1\right ) \left (\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}+\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}\) | \(166\) |
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Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (134) = 268\).
Time = 0.94 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.72 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {3}{10} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )} \log \left (\frac {3}{2} \, \sqrt {2} \sqrt {\sqrt {5} - 5} + \frac {3}{2} \, \sqrt {5} + 6 \, x^{\frac {1}{6}} + \frac {3}{2}\right ) + \frac {3}{10} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} \log \left (-\frac {3}{2} \, \sqrt {2} \sqrt {\sqrt {5} - 5} + \frac {3}{2} \, \sqrt {5} + 6 \, x^{\frac {1}{6}} + \frac {3}{2}\right ) + \frac {1}{10} \, {\left (3 \, \sqrt {5} - \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} - 3\right )} \log \left (-3 \, \sqrt {5} + \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} + 12 \, x^{\frac {1}{6}} + 3\right ) + \frac {1}{10} \, {\left (3 \, \sqrt {5} + \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} - 3\right )} \log \left (-3 \, \sqrt {5} - \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} + 12 \, x^{\frac {1}{6}} + 3\right ) + x + 6 \, x^{\frac {1}{6}} + \frac {6}{5} \, \log \left (x^{\frac {1}{6}} - 1\right ) \]
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\[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=\int \frac {x^{\frac {5}{6}}}{\left (\sqrt [6]{x} - 1\right ) \left (\sqrt [6]{x} + x^{\frac {2}{3}} + \sqrt [3]{x} + \sqrt {x} + 1\right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (134) = 268\).
Time = 0.28 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {3 \, \sqrt {5} \left (-1\right )^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )} \log \left (\frac {\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} - 10} + \left (-1\right )^{\frac {1}{5}} - 4 \, x^{\frac {1}{6}}}{\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} - 10} + \left (-1\right )^{\frac {1}{5}} - 4 \, x^{\frac {1}{6}}}\right )}{5 \, \sqrt {2 \, \sqrt {5} - 10}} - \frac {3 \, \sqrt {5} \left (-1\right )^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )} \log \left (\frac {\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} - 10} - \left (-1\right )^{\frac {1}{5}} + 4 \, x^{\frac {1}{6}}}{\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} - 10} - \left (-1\right )^{\frac {1}{5}} + 4 \, x^{\frac {1}{6}}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} - 10}} - \frac {6}{5} \, \left (-1\right )^{\frac {1}{5}} \log \left (\left (-1\right )^{\frac {1}{5}} + x^{\frac {1}{6}}\right ) + x - \frac {3 \, {\left (\sqrt {5} + 3\right )} \log \left (-x^{\frac {1}{6}} {\left (\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}}\right )} + 2 \, \left (-1\right )^{\frac {2}{5}} + 2 \, x^{\frac {1}{3}}\right )}{5 \, {\left (\sqrt {5} \left (-1\right )^{\frac {4}{5}} + \left (-1\right )^{\frac {4}{5}}\right )}} - \frac {3 \, {\left (\sqrt {5} - 3\right )} \log \left (x^{\frac {1}{6}} {\left (\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}}\right )} + 2 \, \left (-1\right )^{\frac {2}{5}} + 2 \, x^{\frac {1}{3}}\right )}{5 \, {\left (\sqrt {5} \left (-1\right )^{\frac {4}{5}} - \left (-1\right )^{\frac {4}{5}}\right )}} + 6 \, x^{\frac {1}{6}} \]
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Time = 0.44 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {3}{5} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {\sqrt {5} - 4 \, x^{\frac {1}{6}} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) - \frac {3}{5} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {\sqrt {5} + 4 \, x^{\frac {1}{6}} + 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) - \frac {3}{10} \, \sqrt {5} \log \left (\frac {1}{2} \, x^{\frac {1}{6}} {\left (\sqrt {5} + 1\right )} + x^{\frac {1}{3}} + 1\right ) + \frac {3}{10} \, \sqrt {5} \log \left (-\frac {1}{2} \, x^{\frac {1}{6}} {\left (\sqrt {5} - 1\right )} + x^{\frac {1}{3}} + 1\right ) + x + 6 \, x^{\frac {1}{6}} - \frac {3}{10} \, \log \left (x^{\frac {2}{3}} + \sqrt {x} + x^{\frac {1}{3}} + x^{\frac {1}{6}} + 1\right ) + \frac {6}{5} \, \log \left ({\left | x^{\frac {1}{6}} - 1 \right |}\right ) \]
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Time = 17.04 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=x+\frac {6\,\ln \left (1296\,x^{1/6}-1296\right )}{5}-\ln \left (270\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}-270\,\sqrt {5}+1080\,x^{1/6}+270\right )\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}-\frac {3\,\sqrt {5}}{10}+\frac {3}{10}\right )+\ln \left (270\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}+270\,\sqrt {5}-1080\,x^{1/6}-270\right )\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}+\frac {3\,\sqrt {5}}{10}-\frac {3}{10}\right )+6\,x^{1/6}-\ln \left (270\,\sqrt {5}+1080\,x^{1/6}-270\,\sqrt {2}\,\sqrt {\sqrt {5}-5}+270\right )\,\left (\frac {3\,\sqrt {5}}{10}-\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )-\ln \left (270\,\sqrt {5}+1080\,x^{1/6}+270\,\sqrt {2}\,\sqrt {\sqrt {5}-5}+270\right )\,\left (\frac {3\,\sqrt {5}}{10}+\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right ) \]
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