3.32.57 \(\int \frac {1}{3} e^{x^2+\frac {1}{3} (-2+3 e^4+2 x)} (2+6 x) \, dx\)

Optimal. Leaf size=18 \[ e^{e^4+x^2+\frac {1}{3} (-2+2 x)} \]

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Rubi [A]  time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 2244, 2236} \begin {gather*} e^{x^2+\frac {2 x}{3}+\frac {1}{3} \left (3 e^4-2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(x^2 + (-2 + 3*E^4 + 2*x)/3)*(2 + 6*x))/3,x]

[Out]

E^((-2 + 3*E^4)/3 + (2*x)/3 + x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int e^{x^2+\frac {1}{3} \left (-2+3 e^4+2 x\right )} (2+6 x) \, dx\\ &=\frac {1}{3} \int e^{\frac {1}{3} \left (-2+3 e^4\right )+\frac {2 x}{3}+x^2} (2+6 x) \, dx\\ &=e^{\frac {1}{3} \left (-2+3 e^4\right )+\frac {2 x}{3}+x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 17, normalized size = 0.94 \begin {gather*} e^{-\frac {2}{3}+e^4+\frac {2 x}{3}+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x^2 + (-2 + 3*E^4 + 2*x)/3)*(2 + 6*x))/3,x]

[Out]

E^(-2/3 + E^4 + (2*x)/3 + x^2)

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fricas [A]  time = 0.67, size = 11, normalized size = 0.61 \begin {gather*} e^{\left (x^{2} + \frac {2}{3} \, x + e^{4} - \frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(6*x+2)*exp(x^2)*exp(exp(4)+2/3*x-2/3),x, algorithm="fricas")

[Out]

e^(x^2 + 2/3*x + e^4 - 2/3)

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giac [A]  time = 0.39, size = 11, normalized size = 0.61 \begin {gather*} e^{\left (x^{2} + \frac {2}{3} \, x + e^{4} - \frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(6*x+2)*exp(x^2)*exp(exp(4)+2/3*x-2/3),x, algorithm="giac")

[Out]

e^(x^2 + 2/3*x + e^4 - 2/3)

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maple [A]  time = 0.03, size = 12, normalized size = 0.67

method	result	size
gosper	\({\mathrm e}^{{\mathrm e}^{4}+\frac {2 x}{3}-\frac {2}{3}+x^{2}}\)	\(12\)
default	\({\mathrm e}^{{\mathrm e}^{4}+\frac {2 x}{3}-\frac {2}{3}+x^{2}}\)	\(12\)
risch	\({\mathrm e}^{{\mathrm e}^{4}+\frac {2 x}{3}-\frac {2}{3}+x^{2}}\)	\(12\)
norman	\({\mathrm e}^{{\mathrm e}^{4}+\frac {2 x}{3}-\frac {2}{3}} {\mathrm e}^{x^{2}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(6*x+2)*exp(x^2)*exp(exp(4)+2/3*x-2/3),x,method=_RETURNVERBOSE)

[Out]

exp(exp(4)+2/3*x-2/3+x^2)

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maxima [A]  time = 0.39, size = 11, normalized size = 0.61 \begin {gather*} e^{\left (x^{2} + \frac {2}{3} \, x + e^{4} - \frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(6*x+2)*exp(x^2)*exp(exp(4)+2/3*x-2/3),x, algorithm="maxima")

[Out]

e^(x^2 + 2/3*x + e^4 - 2/3)

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mupad [B]  time = 1.88, size = 14, normalized size = 0.78 \begin {gather*} {\mathrm {e}}^{\frac {2\,x}{3}}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-\frac {2}{3}}\,{\mathrm {e}}^{{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*x)/3 + exp(4) - 2/3)*exp(x^2)*(6*x + 2))/3,x)

[Out]

exp((2*x)/3)*exp(x^2)*exp(-2/3)*exp(exp(4))

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sympy [A]  time = 0.60, size = 17, normalized size = 0.94 \begin {gather*} e^{x^{2}} e^{\frac {2 x}{3} - \frac {2}{3} + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(6*x+2)*exp(x**2)*exp(exp(4)+2/3*x-2/3),x)

[Out]

exp(x**2)*exp(2*x/3 - 2/3 + exp(4))

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