### 3.257 $$\int e^{\sqrt{x}} \, dx$$

Optimal. Leaf size=24 $2 e^{\sqrt{x}} \sqrt{x}-2 e^{\sqrt{x}}$

[Out]

-2*E^Sqrt[x] + 2*E^Sqrt[x]*Sqrt[x]

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Rubi [A]  time = 0.0086377, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.429, Rules used = {2207, 2176, 2194} $2 e^{\sqrt{x}} \sqrt{x}-2 e^{\sqrt{x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^Sqrt[x],x]

[Out]

-2*E^Sqrt[x] + 2*E^Sqrt[x]*Sqrt[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]

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

Mathematica [A]  time = 0.0061248, size = 16, normalized size = 0.67 $2 e^{\sqrt{x}} \left (\sqrt{x}-1\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^Sqrt[x],x]

[Out]

2*E^Sqrt[x]*(-1 + Sqrt[x])

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Maple [A]  time = 0., size = 17, normalized size = 0.7 \begin{align*} -2\,{{\rm e}^{\sqrt{x}}}+2\,{{\rm e}^{\sqrt{x}}}\sqrt{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.941537, size = 15, normalized size = 0.62 \begin{align*} 2 \,{\left (\sqrt{x} - 1\right )} e^{\sqrt{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^(1/2)),x, algorithm="maxima")

[Out]

2*(sqrt(x) - 1)*e^sqrt(x)

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Fricas [A]  time = 1.86672, size = 36, normalized size = 1.5 \begin{align*} 2 \,{\left (\sqrt{x} - 1\right )} e^{\sqrt{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^(1/2)),x, algorithm="fricas")

[Out]

2*(sqrt(x) - 1)*e^sqrt(x)

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Sympy [A]  time = 0.175594, size = 20, normalized size = 0.83 \begin{align*} 2 \sqrt{x} e^{\sqrt{x}} - 2 e^{\sqrt{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x**(1/2)),x)

[Out]

2*sqrt(x)*exp(sqrt(x)) - 2*exp(sqrt(x))

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Giac [A]  time = 1.06289, size = 15, normalized size = 0.62 \begin{align*} 2 \,{\left (\sqrt{x} - 1\right )} e^{\sqrt{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^(1/2)),x, algorithm="giac")

[Out]

2*(sqrt(x) - 1)*e^sqrt(x)