Integrand size = 22, antiderivative size = 42 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \arctan \left (\sqrt [6]{x}\right )+3 \log \left (1+\sqrt [3]{x}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6820, 1607, 1816, 649, 209, 266} \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=-6 \arctan \left (\sqrt [6]{x}\right )+\frac {3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]
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Rule 209
Rule 266
Rule 649
Rule 1607
Rule 1816
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+\frac {1}{\sqrt {x}}}{1+\sqrt [3]{x}} \, dx \\ & = 6 \text {Subst}\left (\int \frac {x^2+x^5}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \text {Subst}\left (\int \frac {x^2 \left (1+x^3\right )}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \text {Subst}\left (\int \left (1-x+x^3-\frac {1-x}{1+x^2}\right ) \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \text {Subst}\left (\int \frac {1-x}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [6]{x}\right )+6 \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-6 \tan ^{-1}\left (\sqrt [6]{x}\right )+3 \log \left (1+\sqrt [3]{x}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=\frac {3}{2} \left (4-2 \sqrt [6]{x}+\sqrt {x}\right ) \sqrt [6]{x}-6 \arctan \left (\sqrt [6]{x}\right )+3 \log \left (1+\sqrt [3]{x}\right ) \]
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Time = 1.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(6 x^{\frac {1}{6}}-3 x^{\frac {1}{3}}+\frac {3 x^{\frac {2}{3}}}{2}-6 \arctan \left (x^{\frac {1}{6}}\right )+3 \ln \left (1+x^{\frac {1}{3}}\right )\) | \(31\) |
meijerg | \(6 x^{\frac {1}{6}}-6 \arctan \left (x^{\frac {1}{6}}\right )-\frac {x^{\frac {1}{3}} \left (-3 x^{\frac {1}{3}}+6\right )}{2}+3 \ln \left (1+x^{\frac {1}{3}}\right )\) | \(33\) |
default | \(\ln \left (x +1\right )+\frac {3 x^{\frac {2}{3}}}{2}-\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )+2 \ln \left (1+x^{\frac {1}{3}}\right )-3 x^{\frac {1}{3}}+6 x^{\frac {1}{6}}-6 \arctan \left (x^{\frac {1}{6}}\right )\) | \(48\) |
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Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=\frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
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Time = 4.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \sqrt [6]{x} + \frac {3 x^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x} + 3 \log {\left (\sqrt [3]{x} + 1 \right )} - 6 \operatorname {atan}{\left (\sqrt [6]{x} \right )} \]
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Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=\frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
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Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=\frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
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Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {1+\sqrt {x}}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=\frac {3\,x^{2/3}}{2}+3\,\ln \left (\left (-6+x^{1/6}\,6{}\mathrm {i}\right )\,\left (6+x^{1/6}\,6{}\mathrm {i}\right )\right )-3\,x^{1/3}-6\,\mathrm {atan}\left (x^{1/6}\right )+6\,x^{1/6} \]
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