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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2207, 2176, 2194} \[ 3 e^{\sqrt [3]{x}} x^{2/3}-6 e^{\sqrt [3]{x}} \sqrt [3]{x}+6 e^{\sqrt [3]{x}} \]

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)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2207

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> With[{k = Denominator[n]}, Dist[k/d, Subst[In
t[x^(k - 1)*F^(a + b*x^(k*n)), x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n] &&
!IntegerQ[n]

Rubi steps

\begin {align*} \int e^{\sqrt [3]{x}} \, dx &=3 \operatorname {Subst}\left (\int e^x x^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 e^{\sqrt [3]{x}} x^{2/3}-6 \operatorname {Subst}\left (\int e^x x \, dx,x,\sqrt [3]{x}\right )\\ &=-6 e^{\sqrt [3]{x}} \sqrt [3]{x}+3 e^{\sqrt [3]{x}} x^{2/3}+6 \operatorname {Subst}\left (\int e^x \, dx,x,\sqrt [3]{x}\right )\\ &=6 e^{\sqrt [3]{x}}-6 e^{\sqrt [3]{x}} \sqrt [3]{x}+3 e^{\sqrt [3]{x}} x^{2/3}\\ \end {align*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.40, size = 16, normalized size = 0.42 \[ 3 \, {\left (x^{\frac {2}{3}} - 2 \, x^{\frac {1}{3}} + 2\right )} e^{\left (x^{\frac {1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^(1/3)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.07, size = 16, normalized size = 0.42 \[ 3 \, {\left (x^{\frac {2}{3}} - 2 \, x^{\frac {1}{3}} + 2\right )} e^{\left (x^{\frac {1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^(1/3)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 26, normalized size = 0.68 \[ 3 x^{\frac {2}{3}} {\mathrm e}^{x^{\frac {1}{3}}}-6 x^{\frac {1}{3}} {\mathrm e}^{x^{\frac {1}{3}}}+6 \,{\mathrm e}^{x^{\frac {1}{3}}} \]

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 = 0.57, size = 16, normalized size = 0.42 \[ 3 \, {\left (x^{\frac {2}{3}} - 2 \, x^{\frac {1}{3}} + 2\right )} e^{\left (x^{\frac {1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^(1/3)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 21, normalized size = 0.55 \[ 3\,x\,{\mathrm {e}}^{x^{1/3}}\,\left (\frac {2}{x}+\frac {1}{x^{1/3}}-\frac {2}{x^{2/3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^(1/3)),x)

[Out]

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

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sympy [A]  time = 0.34, 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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