Integrand size = 15, antiderivative size = 41 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=-6 \sqrt [6]{t}+2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+6 \arctan \left (\sqrt [6]{t}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {348, 52, 65, 209} \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=6 \arctan \left (\sqrt [6]{t}\right )+\frac {6 t^{7/6}}{7}-\frac {6 t^{5/6}}{5}+2 \sqrt {t}-6 \sqrt [6]{t} \]
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Rule 52
Rule 65
Rule 209
Rule 348
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {t^{7/2}}{1+t} \, dt,t,\sqrt [3]{t}\right ) \\ & = \frac {6 t^{7/6}}{7}-3 \text {Subst}\left (\int \frac {t^{5/2}}{1+t} \, dt,t,\sqrt [3]{t}\right ) \\ & = -\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+3 \text {Subst}\left (\int \frac {t^{3/2}}{1+t} \, dt,t,\sqrt [3]{t}\right ) \\ & = 2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}-3 \text {Subst}\left (\int \frac {\sqrt {t}}{1+t} \, dt,t,\sqrt [3]{t}\right ) \\ & = -6 \sqrt [6]{t}+2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+3 \text {Subst}\left (\int \frac {1}{\sqrt {t} (1+t)} \, dt,t,\sqrt [3]{t}\right ) \\ & = -6 \sqrt [6]{t}+2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+6 \text {Subst}\left (\int \frac {1}{1+t^2} \, dt,t,\sqrt [6]{t}\right ) \\ & = -6 \sqrt [6]{t}+2 \sqrt {t}-\frac {6 t^{5/6}}{5}+\frac {6 t^{7/6}}{7}+6 \arctan \left (\sqrt [6]{t}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=\frac {2}{35} \left (-105 \sqrt [6]{t}+35 \sqrt {t}-21 t^{5/6}+15 t^{7/6}\right )+6 \arctan \left (\sqrt [6]{t}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(-6 t^{\frac {1}{6}}-\frac {6 t^{\frac {5}{6}}}{5}+\frac {6 t^{\frac {7}{6}}}{7}+6 \arctan \left (t^{\frac {1}{6}}\right )+2 \sqrt {t}\) | \(28\) |
default | \(-6 t^{\frac {1}{6}}-\frac {6 t^{\frac {5}{6}}}{5}+\frac {6 t^{\frac {7}{6}}}{7}+6 \arctan \left (t^{\frac {1}{6}}\right )+2 \sqrt {t}\) | \(28\) |
meijerg | \(-\frac {2 t^{\frac {1}{6}} \left (-45 t +63 t^{\frac {2}{3}}-105 t^{\frac {1}{3}}+315\right )}{105}+6 \arctan \left (t^{\frac {1}{6}}\right )\) | \(28\) |
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Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=\frac {6}{7} \, {\left (t - 7\right )} t^{\frac {1}{6}} - \frac {6}{5} \, t^{\frac {5}{6}} + 2 \, \sqrt {t} + 6 \, \arctan \left (t^{\frac {1}{6}}\right ) \]
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Time = 1.70 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=\frac {6 t^{\frac {7}{6}}}{7} - \frac {6 t^{\frac {5}{6}}}{5} - 6 \sqrt [6]{t} + 2 \sqrt {t} + 6 \operatorname {atan}{\left (\sqrt [6]{t} \right )} \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=\frac {6}{7} \, t^{\frac {7}{6}} - \frac {6}{5} \, t^{\frac {5}{6}} + 2 \, \sqrt {t} - 6 \, t^{\frac {1}{6}} + 6 \, \arctan \left (t^{\frac {1}{6}}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=\frac {6}{7} \, t^{\frac {7}{6}} - \frac {6}{5} \, t^{\frac {5}{6}} + 2 \, \sqrt {t} - 6 \, t^{\frac {1}{6}} + 6 \, \arctan \left (t^{\frac {1}{6}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {t}}{1+\sqrt [3]{t}} \, dt=6\,\mathrm {atan}\left (t^{1/6}\right )+2\,\sqrt {t}-6\,t^{1/6}-\frac {6\,t^{5/6}}{5}+\frac {6\,t^{7/6}}{7} \]
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