Optimal. Leaf size=41 \[ \frac{6 t^{7/6}}{7}-\frac{6 t^{5/6}}{5}+2 \sqrt{t}-6 \sqrt [6]{t}+6 \tan ^{-1}\left (\sqrt [6]{t}\right ) \]
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Rubi [A] time = 0.0119473, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {341, 50, 63, 203} \[ \frac{6 t^{7/6}}{7}-\frac{6 t^{5/6}}{5}+2 \sqrt{t}-6 \sqrt [6]{t}+6 \tan ^{-1}\left (\sqrt [6]{t}\right ) \]
Antiderivative was successfully verified.
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Rule 341
Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt{t}}{1+\sqrt [3]{t}} \, dt &=3 \operatorname{Subst}\left (\int \frac{t^{7/2}}{1+t} \, dt,t,\sqrt [3]{t}\right )\\ &=\frac{6 t^{7/6}}{7}-3 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \tan ^{-1}\left (\sqrt [6]{t}\right )\\ \end{align*}
Mathematica [A] time = 0.0095159, size = 41, normalized size = 1. \[ \frac{6 t^{7/6}}{7}-\frac{6 t^{5/6}}{5}+2 \sqrt{t}-6 \sqrt [6]{t}+6 \tan ^{-1}\left (\sqrt [6]{t}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 28, normalized size = 0.7 \begin{align*} -6\,\sqrt [6]{t}-{\frac{6}{5}{t}^{{\frac{5}{6}}}}+{\frac{6}{7}{t}^{{\frac{7}{6}}}}+6\,\arctan \left ( \sqrt [6]{t} \right ) +2\,\sqrt{t} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40737, size = 36, normalized size = 0.88 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12846, size = 90, normalized size = 2.2 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.93056, size = 37, normalized size = 0.9 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05334, size = 36, normalized size = 0.88 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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