3.4.7 \(\int e^{\sqrt [3]{x}} \, dx\) [307]

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

[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 = 0.01, 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 = {2238, 2207, 2225} \begin {gather*} 3 e^{\sqrt [3]{x}} x^{2/3}-6 e^{\sqrt [3]{x}} \sqrt [3]{x}+6 e^{\sqrt [3]{x}} \end {gather*}

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 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 2238

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

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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Mathics [A]
time = 1.89, size = 17, normalized size = 0.45 \begin {gather*} 3 \left (2-2 x^{\frac {1}{3}}+x^{\frac {2}{3}}\right ) E^{x^{\frac {1}{3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[E^x^(1/3),x]')

[Out]

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

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Maple [A]
time = 0.01, size = 26, normalized size = 0.68

method	result	size
meijerg	\(-6+\left (3 x^{\frac {2}{3}}-6 x^{\frac {1}{3}}+6\right ) {\mathrm e}^{x^{\frac {1}{3}}}\)	\(20\)
derivativedivides	\(6 \,{\mathrm e}^{x^{\frac {1}{3}}}-6 \,{\mathrm e}^{x^{\frac {1}{3}}} x^{\frac {1}{3}}+3 \,{\mathrm e}^{x^{\frac {1}{3}}} x^{\frac {2}{3}}\)	\(26\)
default	\(6 \,{\mathrm e}^{x^{\frac {1}{3}}}-6 \,{\mathrm e}^{x^{\frac {1}{3}}} x^{\frac {1}{3}}+3 \,{\mathrm e}^{x^{\frac {1}{3}}} x^{\frac {2}{3}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^(1/3)),x,method=_RETURNVERBOSE)

[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 = 0.27, size = 16, normalized size = 0.42 \begin {gather*} 3 \, {\left (x^{\frac {2}{3}} - 2 \, x^{\frac {1}{3}} + 2\right )} e^{\left (x^{\frac {1}{3}}\right )} \end {gather*}

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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Fricas [A]
time = 0.34, size = 16, normalized size = 0.42 \begin {gather*} 3 \, {\left (x^{\frac {2}{3}} - 2 \, x^{\frac {1}{3}} + 2\right )} e^{\left (x^{\frac {1}{3}}\right )} \end {gather*}

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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Sympy [A]
time = 0.11, size = 34, normalized size = 0.89 \begin {gather*} 3 x^{\frac {2}{3}} e^{\sqrt [3]{x}} - 6 \sqrt [3]{x} e^{\sqrt [3]{x}} + 6 e^{\sqrt [3]{x}} \end {gather*}

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 [A]
time = 0.00, size = 25, normalized size = 0.66 \begin {gather*} 3 \left (\left (x^{\frac {1}{3}}\right )^{2}-2 x^{\frac {1}{3}}+2\right ) \mathrm {e}^{x^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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