### 3.307 $$\int e^{\sqrt {x}} \, dx$$

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

[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 = 0.018782, 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, Rules used = {2207, 2176, 2194} $3 e^{\sqrt {x}} x^{2/3}-6 e^{\sqrt {x}} \sqrt {x}+6 e^{\sqrt {x}}$

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 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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0080055, size = 24, normalized size = 0.63 $e^{\sqrt {x}} \left (3 x^{2/3}-6 \sqrt {x}+6\right )$

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 = 0.002, size = 26, normalized size = 0.7 \begin{align*} 6\,{{\rm e}^{\sqrt {x}}}-6\,{{\rm e}^{\sqrt {x}}}\sqrt {x}+3\,{{\rm e}^{\sqrt {x}}}{x}^{2/3} \end{align*}

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 = 0.936532, size = 22, normalized size = 0.58 \begin{align*} 3 \,{\left (x^{\frac{2}{3}} - 2 \, x^{\frac{1}{3}} + 2\right )} e^{\left (x^{\frac{1}{3}}\right )} \end{align*}

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 = 2.19803, size = 55, normalized size = 1.45 \begin{align*} 3 \,{\left (x^{\frac{2}{3}} - 2 \, x^{\frac{1}{3}} + 2\right )} e^{\left (x^{\frac{1}{3}}\right )} \end{align*}

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 = 0.318136, size = 34, normalized size = 0.89 \begin{align*} 3 x^{\frac{2}{3}} e^{\sqrt {x}} - 6 \sqrt {x} e^{\sqrt {x}} + 6 e^{\sqrt {x}} \end{align*}

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 = 1.05636, size = 22, normalized size = 0.58 \begin{align*} 3 \,{\left (x^{\frac{2}{3}} - 2 \, x^{\frac{1}{3}} + 2\right )} e^{\left (x^{\frac{1}{3}}\right )} \end{align*}

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))