3.1.78 \(\int e^{\sqrt {x}} \, dx\) [78]

Optimal. Leaf size=24 \[ -2 e^{\sqrt {x}}+2 e^{\sqrt {x}} \sqrt {x} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {2238, 2207, 2225} \begin {gather*} 2 e^{\sqrt {x}} \sqrt {x}-2 e^{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^Sqrt[x],x]

[Out]

-2*E^Sqrt[x] + 2*E^Sqrt[x]*Sqrt[x]

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

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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.67 \begin {gather*} 2 e^{\sqrt {x}} \left (-1+\sqrt {x}\right ) \end {gather*}

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 = 0.00, size = 17, normalized size = 0.71

method	result	size
meijerg	\(2-\left (-2 \sqrt {x}+2\right ) {\mathrm e}^{\sqrt {x}}\)	\(16\)
derivativedivides	\(-2 \,{\mathrm e}^{\sqrt {x}}+2 \,{\mathrm e}^{\sqrt {x}} \sqrt {x}\)	\(17\)
default	\(-2 \,{\mathrm e}^{\sqrt {x}}+2 \,{\mathrm e}^{\sqrt {x}} \sqrt {x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.81, size = 11, normalized size = 0.46 \begin {gather*} 2 \, {\left (\sqrt {x} - 1\right )} e^{\sqrt {x}} \end {gather*}

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 = 0.88, size = 11, normalized size = 0.46 \begin {gather*} 2 \, {\left (\sqrt {x} - 1\right )} e^{\sqrt {x}} \end {gather*}

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 = 0.06, size = 20, normalized size = 0.83 \begin {gather*} 2 \sqrt {x} e^{\sqrt {x}} - 2 e^{\sqrt {x}} \end {gather*}

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 = 0.51, size = 11, normalized size = 0.46 \begin {gather*} 2 \, {\left (\sqrt {x} - 1\right )} e^{\sqrt {x}} \end {gather*}

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)

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Mupad [B]
time = 0.02, size = 11, normalized size = 0.46 \begin {gather*} 2\,{\mathrm {e}}^{\sqrt {x}}\,\left (\sqrt {x}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*exp(x^(1/2))*(x^(1/2) - 1)

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