Integrand size = 15, antiderivative size = 16 \[ \int \frac {1}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \sqrt [6]{x}-6 \arctan \left (\sqrt [6]{x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {348, 52, 65, 209} \[ \int \frac {1}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \sqrt [6]{x}-6 \arctan \left (\sqrt [6]{x}\right ) \]
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Rule 52
Rule 65
Rule 209
Rule 348
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {\sqrt {x}}{1+x} \, dx,x,\sqrt [3]{x}\right ) \\ & = 6 \sqrt [6]{x}-3 \text {Subst}\left (\int \frac {1}{\sqrt {x} (1+x)} \, dx,x,\sqrt [3]{x}\right ) \\ & = 6 \sqrt [6]{x}-6 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}-6 \arctan \left (\sqrt [6]{x}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \sqrt [6]{x}-6 \arctan \left (\sqrt [6]{x}\right ) \]
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Time = 0.16 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(6 x^{\frac {1}{6}}-6 \arctan \left (x^{\frac {1}{6}}\right )\) | \(13\) |
default | \(6 x^{\frac {1}{6}}-6 \arctan \left (x^{\frac {1}{6}}\right )\) | \(13\) |
meijerg | \(6 x^{\frac {1}{6}}-6 \arctan \left (x^{\frac {1}{6}}\right )\) | \(13\) |
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Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) \]
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Time = 0.41 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \sqrt [6]{x} + 6 \operatorname {atan}{\left (\frac {1}{\sqrt [6]{x}} \right )} \]
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Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6 \, x^{\frac {1}{6}} - 6 \, \arctan \left (x^{\frac {1}{6}}\right ) \]
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Time = 14.48 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (1+\sqrt [3]{x}\right ) \sqrt {x}} \, dx=6\,x^{1/6}-6\,\mathrm {atan}\left (x^{1/6}\right ) \]
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