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

Optimal. Leaf size=26 \[ x+e^{\frac {2 e^2 (4-x) x}{3 (-4+x)}} x^2 \]

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Rubi [A]  time = 0.09, antiderivative size = 16, normalized size of antiderivative = 0.62, number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {12, 1593, 2196, 2176, 2194} \begin {gather*} e^{-\frac {2 e^2 x}{3}} x^2+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + x^2/E^((2*E^2*x)/3)

Rule 12

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (3+e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right )\right ) \, dx\\ &=x+\frac {1}{3} \int e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right ) \, dx\\ &=x+\frac {1}{3} \int e^{-\frac {2 e^2 x}{3}} x \left (6-2 e^2 x\right ) \, dx\\ &=x+\frac {1}{3} \int \left (6 e^{-\frac {2 e^2 x}{3}} x-2 e^{2-\frac {2 e^2 x}{3}} x^2\right ) \, dx\\ &=x-\frac {2}{3} \int e^{2-\frac {2 e^2 x}{3}} x^2 \, dx+2 \int e^{-\frac {2 e^2 x}{3}} x \, dx\\ &=x-3 e^{-2-\frac {2 e^2 x}{3}} x+e^{-\frac {2 e^2 x}{3}} x^2-\frac {2 \int e^{2-\frac {2 e^2 x}{3}} x \, dx}{e^2}+\frac {3 \int e^{-\frac {2 e^2 x}{3}} \, dx}{e^2}\\ &=-\frac {9}{2} e^{-4-\frac {2 e^2 x}{3}}+x+e^{-\frac {2 e^2 x}{3}} x^2-\frac {3 \int e^{2-\frac {2 e^2 x}{3}} \, dx}{e^4}\\ &=x+e^{-\frac {2 e^2 x}{3}} x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.62 \begin {gather*} x+e^{-\frac {2 e^2 x}{3}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + x^2/E^((2*E^2*x)/3)

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fricas [A]  time = 0.55, size = 12, normalized size = 0.46 \begin {gather*} x^{2} e^{\left (-\frac {2}{3} \, x e^{2}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^(-2/3*x*e^2) + x

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giac [B]  time = 0.24, size = 43, normalized size = 1.65 \begin {gather*} \frac {1}{2} \, {\left (2 \, x^{2} e^{4} + 6 \, x e^{2} + 9\right )} e^{\left (-\frac {2}{3} \, x e^{2} - 4\right )} - \frac {3}{2} \, {\left (2 \, x e^{2} + 3\right )} e^{\left (-\frac {2}{3} \, x e^{2} - 4\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*x^2*e^4 + 6*x*e^2 + 9)*e^(-2/3*x*e^2 - 4) - 3/2*(2*x*e^2 + 3)*e^(-2/3*x*e^2 - 4) + x

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maple [A]  time = 0.04, size = 13, normalized size = 0.50

method	result	size
risch	\(x +x^{2} {\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}\)	\(13\)
norman	\(x +x^{2} {\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}\)	\(15\)
default	\(x -18 \,{\mathrm e}^{-2} \left (-{\mathrm e}^{-2} \left (-\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}} {\mathrm e}^{2} x}{6}-\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{4}\right )-{\mathrm e}^{-2} \left (\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}} {\mathrm e}^{4} x^{2}}{18}+\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}} {\mathrm e}^{2} x}{6}+\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{4}\right )\right )\)	\(87\)
derivativedivides	\(-{\mathrm e}^{-2} \left (-{\mathrm e}^{2} x -18 \,{\mathrm e}^{-2} \left (-\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}} {\mathrm e}^{2} x}{6}-\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{4}\right )-18 \,{\mathrm e}^{-2} \left (\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}} {\mathrm e}^{4} x^{2}}{18}+\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}} {\mathrm e}^{2} x}{6}+\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{4}\right )\right )\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.36, size = 43, normalized size = 1.65 \begin {gather*} \frac {1}{2} \, {\left (2 \, x^{2} e^{4} + 6 \, x e^{2} + 9\right )} e^{\left (-\frac {2}{3} \, x e^{2} - 4\right )} - \frac {3}{2} \, {\left (2 \, x e^{2} + 3\right )} e^{\left (-\frac {2}{3} \, x e^{2} - 4\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*x^2*e^4 + 6*x*e^2 + 9)*e^(-2/3*x*e^2 - 4) - 3/2*(2*x*e^2 + 3)*e^(-2/3*x*e^2 - 4) + x

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mupad [B]  time = 0.06, size = 12, normalized size = 0.46 \begin {gather*} x+x^2\,{\mathrm {e}}^{-\frac {2\,x\,{\mathrm {e}}^2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(2*x*exp(2))/3)*(6*x - 2*x^2*exp(2)))/3 + 1,x)

[Out]

x + x^2*exp(-(2*x*exp(2))/3)

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sympy [A]  time = 0.11, size = 14, normalized size = 0.54 \begin {gather*} x^{2} e^{- \frac {2 x e^{2}}{3}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*exp(-2*x*exp(2)/3) + x

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