3.319 \(\int \frac{\sqrt{t}}{1+\sqrt [3]{t}} \, dt\)

Optimal. Leaf size=41 \[ \frac{6 t^{7/6}}{7}-\frac{6 t^{5/6}}{5}+2 \sqrt{t}-6 \sqrt [6]{t}+6 \tan ^{-1}\left (\sqrt [6]{t}\right ) \]

[Out]

-6*t^(1/6) + 2*Sqrt[t] - (6*t^(5/6))/5 + (6*t^(7/6))/7 + 6*ArcTan[t^(1/6)]

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Rubi [A]  time = 0.033075, 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 \[ \frac{6 t^{7/6}}{7}-\frac{6 t^{5/6}}{5}+2 \sqrt{t}-6 \sqrt [6]{t}+6 \tan ^{-1}\left (\sqrt [6]{t}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[t]/(1 + t^(1/3)),t]

[Out]

-6*t^(1/6) + 2*Sqrt[t] - (6*t^(5/6))/5 + (6*t^(7/6))/7 + 6*ArcTan[t^(1/6)]

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Rubi in Sympy [A]  time = 2.71242, size = 37, normalized size = 0.9 \[ \frac{6 t^{\frac{7}{6}}}{7} - \frac{6 t^{\frac{5}{6}}}{5} - 6 \sqrt [6]{t} + 2 \sqrt{t} + 6 \operatorname{atan}{\left (\sqrt [6]{t} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(t**(1/2)/(1+t**(1/3)),t)

[Out]

6*t**(7/6)/7 - 6*t**(5/6)/5 - 6*t**(1/6) + 2*sqrt(t) + 6*atan(t**(1/6))

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Mathematica [A]  time = 0.0144812, 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.

[In]  Integrate[Sqrt[t]/(1 + t^(1/3)),t]

[Out]

-6*t^(1/6) + 2*Sqrt[t] - (6*t^(5/6))/5 + (6*t^(7/6))/7 + 6*ArcTan[t^(1/6)]

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Maple [A]  time = 0.004, size = 28, normalized size = 0.7 \[ -6\,\sqrt [6]{t}-{\frac{6}{5}{t}^{{\frac{5}{6}}}}+{\frac{6}{7}{t}^{{\frac{7}{6}}}}+6\,\arctan \left ( \sqrt [6]{t} \right ) +2\,\sqrt{t} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(t^(1/2)/(1+t^(1/3)),t)

[Out]

-6*t^(1/6)-6/5*t^(5/6)+6/7*t^(7/6)+6*arctan(t^(1/6))+2*t^(1/2)

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Maxima [A]  time = 1.49266, size = 36, normalized size = 0.88 \[ \frac{6}{7} \, t^{\frac{7}{6}} - \frac{6}{5} \, t^{\frac{5}{6}} + 2 \, \sqrt{t} - 6 \, t^{\frac{1}{6}} + 6 \, \arctan \left (t^{\frac{1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(t)/(t^(1/3) + 1),t, algorithm="maxima")

[Out]

6/7*t^(7/6) - 6/5*t^(5/6) + 2*sqrt(t) - 6*t^(1/6) + 6*arctan(t^(1/6))

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Fricas [A]  time = 0.215419, size = 34, normalized size = 0.83 \[ \frac{6}{7} \,{\left (t - 7\right )} t^{\frac{1}{6}} - \frac{6}{5} \, t^{\frac{5}{6}} + 2 \, \sqrt{t} + 6 \, \arctan \left (t^{\frac{1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(t)/(t^(1/3) + 1),t, algorithm="fricas")

[Out]

6/7*(t - 7)*t^(1/6) - 6/5*t^(5/6) + 2*sqrt(t) + 6*arctan(t^(1/6))

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Sympy [A]  time = 4.80965, size = 37, normalized size = 0.9 \[ \frac{6 t^{\frac{7}{6}}}{7} - \frac{6 t^{\frac{5}{6}}}{5} - 6 \sqrt [6]{t} + 2 \sqrt{t} + 6 \operatorname{atan}{\left (\sqrt [6]{t} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t**(1/2)/(1+t**(1/3)),t)

[Out]

6*t**(7/6)/7 - 6*t**(5/6)/5 - 6*t**(1/6) + 2*sqrt(t) + 6*atan(t**(1/6))

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GIAC/XCAS [A]  time = 0.213486, size = 36, normalized size = 0.88 \[ \frac{6}{7} \, t^{\frac{7}{6}} - \frac{6}{5} \, t^{\frac{5}{6}} + 2 \, \sqrt{t} - 6 \, t^{\frac{1}{6}} + 6 \, \arctan \left (t^{\frac{1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(t)/(t^(1/3) + 1),t, algorithm="giac")

[Out]

6/7*t^(7/6) - 6/5*t^(5/6) + 2*sqrt(t) - 6*t^(1/6) + 6*arctan(t^(1/6))