Optimal. Leaf size=201 \[ x+6 \sqrt [6]{x}+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt {5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt {5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {4 \sqrt [6]{x}-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (4 \sqrt [6]{x}+\sqrt {5}+1\right )\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1584, 341, 302, 202, 634, 618, 204, 628, 31} \begin {gather*} 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 )-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\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 )} \tan ^{-1}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (4 \sqrt [6]{x}+\sqrt {5}+1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 202
Rule 204
Rule 302
Rule 341
Rule 618
Rule 628
Rule 634
Rule 1584
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx &=\int \frac {x^{5/6}}{-1+x^{5/6}} \, dx\\ &=6 \operatorname {Subst}\left (\int \frac {x^{10}}{-1+x^5} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname {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 \operatorname {Subst}\left (\int \frac {1}{-1+x^5} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}+x-\frac {6}{5} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [6]{x}\right )-\frac {12}{5} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 0.14 \begin {gather*} -6 \sqrt [6]{x} \, _2F_1\left (\frac {1}{5},1;\frac {6}{5};x^{5/6}\right )+x+6 \sqrt [6]{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.08, size = 127, normalized size = 0.63 \begin {gather*} -\frac {6}{5} \text {RootSum}\left [\text {$\#$1}^4+\text {$\#$1}^3+\text {$\#$1}^2+\text {$\#$1}+1\&,\frac {\text {$\#$1}^3 \log \left (\sqrt [6]{x}-\text {$\#$1}\right )+2 \text {$\#$1}^2 \log \left (\sqrt [6]{x}-\text {$\#$1}\right )+3 \text {$\#$1} \log \left (\sqrt [6]{x}-\text {$\#$1}\right )+4 \log \left (\sqrt [6]{x}-\text {$\#$1}\right )}{4 \text {$\#$1}^3+3 \text {$\#$1}^2+2 \text {$\#$1}+1}\&\right ]+x+6 \sqrt [6]{x}+\frac {6}{5} \log \left (\sqrt [6]{x}-1\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.83, size = 547, normalized size = 2.72 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 140, normalized size = 0.70 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 242, normalized size = 1.20 \begin {gather*} x -\frac {6 \arctan \left (\frac {4 x^{\frac {1}{6}}+1-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{\sqrt {10+2 \sqrt {5}}}-\frac {6 \sqrt {5}\, \arctan \left (\frac {4 x^{\frac {1}{6}}+1-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {6 \arctan \left (\frac {4 x^{\frac {1}{6}}+1+\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{\sqrt {10-2 \sqrt {5}}}+\frac {6 \sqrt {5}\, \arctan \left (\frac {4 x^{\frac {1}{6}}+1+\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {6 \ln \left (x^{\frac {1}{6}}-1\right )}{5}+\frac {3 \sqrt {5}\, \ln \left (2 x^{\frac {1}{3}}+x^{\frac {1}{6}}-\sqrt {5}\, x^{\frac {1}{6}}+2\right )}{10}-\frac {3 \ln \left (2 x^{\frac {1}{3}}+x^{\frac {1}{6}}-\sqrt {5}\, x^{\frac {1}{6}}+2\right )}{10}-\frac {3 \sqrt {5}\, \ln \left (2 x^{\frac {1}{3}}+x^{\frac {1}{6}}+\sqrt {5}\, x^{\frac {1}{6}}+2\right )}{10}-\frac {3 \ln \left (2 x^{\frac {1}{3}}+x^{\frac {1}{6}}+\sqrt {5}\, x^{\frac {1}{6}}+2\right )}{10}+6 x^{\frac {1}{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.52, size = 293, normalized size = 1.46 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 208, normalized size = 1.03 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.15, size = 311, normalized size = 1.55 \begin {gather*} 6 \sqrt [6]{x} + x + \frac {6 \log {\left (\sqrt [6]{x} - 1 \right )}}{5} - \frac {3 \sqrt {5} \log {\left (8 \sqrt [6]{x} + 8 \sqrt {5} \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} - \frac {3 \log {\left (8 \sqrt [6]{x} + 8 \sqrt {5} \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} - \frac {3 \log {\left (- 8 \sqrt {5} \sqrt [6]{x} + 8 \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} + \frac {3 \sqrt {5} \log {\left (- 8 \sqrt {5} \sqrt [6]{x} + 8 \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} - \frac {3 \sqrt {2} \sqrt {5 - \sqrt {5}} \operatorname {atan}{\left (\frac {2 \sqrt {2} \sqrt [6]{x}}{\sqrt {5 - \sqrt {5}}} + \frac {\sqrt {2}}{2 \sqrt {5 - \sqrt {5}}} + \frac {\sqrt {10}}{2 \sqrt {5 - \sqrt {5}}} \right )}}{5} - \frac {3 \sqrt {2} \sqrt {\sqrt {5} + 5} \operatorname {atan}{\left (\frac {2 \sqrt {2} \sqrt [6]{x}}{\sqrt {\sqrt {5} + 5}} - \frac {\sqrt {10}}{2 \sqrt {\sqrt {5} + 5}} + \frac {\sqrt {2}}{2 \sqrt {\sqrt {5} + 5}} \right )}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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