3.307 \(\int e^{\sqrt [3]{x}} \, dx\)

Optimal. Leaf size=38 \[ 3 e^{\sqrt [3]{x}} x^{2/3}-6 e^{\sqrt [3]{x}} \sqrt [3]{x}+6 e^{\sqrt [3]{x}} \]

[Out]

6*E^x^(1/3) - 6*E^x^(1/3)*x^(1/3) + 3*E^x^(1/3)*x^(2/3)

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Rubi [A]  time = 0.0325605, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ 3 e^{\sqrt [3]{x}} x^{2/3}-6 e^{\sqrt [3]{x}} \sqrt [3]{x}+6 e^{\sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]  Int[E^x^(1/3),x]

[Out]

6*E^x^(1/3) - 6*E^x^(1/3)*x^(1/3) + 3*E^x^(1/3)*x^(2/3)

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Rubi in Sympy [A]  time = 1.94994, size = 34, normalized size = 0.89 \[ 3 x^{\frac{2}{3}} e^{\sqrt [3]{x}} - 6 \sqrt [3]{x} e^{\sqrt [3]{x}} + 6 e^{\sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x**(1/3)),x)

[Out]

3*x**(2/3)*exp(x**(1/3)) - 6*x**(1/3)*exp(x**(1/3)) + 6*exp(x**(1/3))

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Mathematica [A]  time = 0.00603552, size = 24, normalized size = 0.63 \[ e^{\sqrt [3]{x}} \left (3 x^{2/3}-6 \sqrt [3]{x}+6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x^(1/3),x]

[Out]

E^x^(1/3)*(6 - 6*x^(1/3) + 3*x^(2/3))

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Maple [A]  time = 0.003, size = 26, normalized size = 0.7 \[ 6\,{{\rm e}^{\sqrt [3]{x}}}-6\,{{\rm e}^{\sqrt [3]{x}}}\sqrt [3]{x}+3\,{{\rm e}^{\sqrt [3]{x}}}{x}^{2/3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x^(1/3)),x)

[Out]

6*exp(x^(1/3))-6*exp(x^(1/3))*x^(1/3)+3*exp(x^(1/3))*x^(2/3)

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Maxima [A]  time = 1.37396, size = 22, normalized size = 0.58 \[ 3 \,{\left (x^{\frac{2}{3}} - 2 \, x^{\frac{1}{3}} + 2\right )} e^{\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(x^(1/3)),x, algorithm="maxima")

[Out]

3*(x^(2/3) - 2*x^(1/3) + 2)*e^(x^(1/3))

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Fricas [A]  time = 0.231071, size = 22, normalized size = 0.58 \[ 3 \,{\left (x^{\frac{2}{3}} - 2 \, x^{\frac{1}{3}} + 2\right )} e^{\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(x^(1/3)),x, algorithm="fricas")

[Out]

3*(x^(2/3) - 2*x^(1/3) + 2)*e^(x^(1/3))

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Sympy [A]  time = 0.438435, size = 34, normalized size = 0.89 \[ 3 x^{\frac{2}{3}} e^{\sqrt [3]{x}} - 6 \sqrt [3]{x} e^{\sqrt [3]{x}} + 6 e^{\sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x**(1/3)),x)

[Out]

3*x**(2/3)*exp(x**(1/3)) - 6*x**(1/3)*exp(x**(1/3)) + 6*exp(x**(1/3))

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GIAC/XCAS [A]  time = 0.213582, size = 22, normalized size = 0.58 \[ 3 \,{\left (x^{\frac{2}{3}} - 2 \, x^{\frac{1}{3}} + 2\right )} e^{\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(x^(1/3)),x, algorithm="giac")

[Out]

3*(x^(2/3) - 2*x^(1/3) + 2)*e^(x^(1/3))