Integrand size = 15, antiderivative size = 108 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=2 \sqrt {x}+\frac {3 \arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \arctan \left (1+\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}} \]
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Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1598, 348, 327, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=\frac {3 \arctan \left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \arctan \left (\sqrt {2} \sqrt [6]{x}+1\right )}{\sqrt {2}}+2 \sqrt {x}-\frac {3 \log \left (\sqrt [3]{x}-\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}}+\frac {3 \log \left (\sqrt [3]{x}+\sqrt {2} \sqrt [6]{x}+1\right )}{2 \sqrt {2}} \]
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Rule 210
Rule 303
Rule 327
Rule 335
Rule 348
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [6]{x}}{1+x^{2/3}} \, dx \\ & = 3 \text {Subst}\left (\int \frac {x^{5/2}}{1+x^2} \, dx,x,\sqrt [3]{x}\right ) \\ & = 2 \sqrt {x}-3 \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\sqrt [3]{x}\right ) \\ & = 2 \sqrt {x}-6 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right ) \\ & = 2 \sqrt {x}+3 \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )-3 \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right ) \\ & = 2 \sqrt {x}-\frac {3}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {3 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [6]{x}\right )}{2 \sqrt {2}}-\frac {3 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [6]{x}\right )}{2 \sqrt {2}} \\ & = 2 \sqrt {x}-\frac {3 \log \left (1-\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}}-\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}+\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}} \\ & = 2 \sqrt {x}+\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \tan ^{-1}\left (1+\sqrt {2} \sqrt [6]{x}\right )}{\sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt {2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=2 \sqrt {x}-\frac {3 \arctan \left (\frac {-1+\sqrt [3]{x}}{\sqrt {2} \sqrt [6]{x}}\right )}{\sqrt {2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [6]{x}}{1+\sqrt [3]{x}}\right )}{\sqrt {2}} \]
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Time = 0.95 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(2 \sqrt {x}-\frac {3 \sqrt {2}\, \left (\ln \left (\frac {1+x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {2}}{1+x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {2}}\right )+2 \arctan \left (1+x^{\frac {1}{6}} \sqrt {2}\right )+2 \arctan \left (-1+x^{\frac {1}{6}} \sqrt {2}\right )\right )}{4}\) | \(66\) |
default | \(2 \sqrt {x}-\frac {3 \sqrt {2}\, \left (\ln \left (\frac {1+x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {2}}{1+x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {2}}\right )+2 \arctan \left (1+x^{\frac {1}{6}} \sqrt {2}\right )+2 \arctan \left (-1+x^{\frac {1}{6}} \sqrt {2}\right )\right )}{4}\) | \(66\) |
meijerg | \(2 \sqrt {x}-\frac {3 \sqrt {x}\, \left (\frac {\sqrt {2}\, \ln \left (1+x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {2}\right )}{2 \sqrt {x}}+\frac {\sqrt {2}\, \arctan \left (\frac {x^{\frac {1}{6}} \sqrt {2}}{2-x^{\frac {1}{6}} \sqrt {2}}\right )}{\sqrt {x}}-\frac {\sqrt {2}\, \ln \left (1+x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {2}\right )}{2 \sqrt {x}}+\frac {\sqrt {2}\, \arctan \left (\frac {x^{\frac {1}{6}} \sqrt {2}}{2+x^{\frac {1}{6}} \sqrt {2}}\right )}{\sqrt {x}}\right )}{2}\) | \(112\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=-\left (\frac {3}{4} i - \frac {3}{4}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) + \left (\frac {3}{4} i + \frac {3}{4}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) - \left (\frac {3}{4} i + \frac {3}{4}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) + \left (\frac {3}{4} i - \frac {3}{4}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, x^{\frac {1}{6}}\right ) + 2 \, \sqrt {x} \]
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\[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=\int \frac {\sqrt {x}}{\sqrt [3]{x} + x}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=-\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, x^{\frac {1}{6}}\right )}\right ) - \frac {3}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + 2 \, \sqrt {x} \]
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Time = 0.35 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=-\frac {3}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, x^{\frac {1}{6}}\right )}\right ) - \frac {3}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, x^{\frac {1}{6}}\right )}\right ) + \frac {3}{4} \, \sqrt {2} \log \left (\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{4} \, \sqrt {2} \log \left (-\sqrt {2} x^{\frac {1}{6}} + x^{\frac {1}{3}} + 1\right ) + 2 \, \sqrt {x} \]
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Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {x}}{\sqrt [3]{x}+x} \, dx=2\,\sqrt {x}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^{1/6}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {3}{2}+\frac {3}{2}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^{1/6}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {3}{2}-\frac {3}{2}{}\mathrm {i}\right ) \]
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