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

Optimal. Leaf size=22 \[ x+x \left (x-\frac {1}{2} \left (e^{-6+2 x}+\frac {1}{x}\right ) x\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {12, 1593, 2196, 2176, 2194} \begin {gather*} -\frac {1}{2} e^{2 x-6} x^2+x^2+\frac {x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 4*x + E^(-6 + 2*x)*(-2*x - 2*x^2))/2,x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 23, normalized size = 1.05 \begin {gather*} \frac {x}{2}+x^2-\frac {1}{2} e^{-6+2 x} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 4*x + E^(-6 + 2*x)*(-2*x - 2*x^2))/2,x]

[Out]

x/2 + x^2 - (E^(-6 + 2*x)*x^2)/2

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fricas [A]  time = 0.79, size = 18, normalized size = 0.82 \begin {gather*} -\frac {1}{2} \, x^{2} e^{\left (2 \, x - 6\right )} + x^{2} + \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^2*e^(2*x - 6) + x^2 + 1/2*x

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giac [A]  time = 0.24, size = 18, normalized size = 0.82 \begin {gather*} -\frac {1}{2} \, x^{2} e^{\left (2 \, x - 6\right )} + x^{2} + \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^2*e^(2*x - 6) + x^2 + 1/2*x

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maple [A]  time = 0.05, size = 19, normalized size = 0.86

method	result	size
norman	\(x^{2}+\frac {x}{2}-\frac {{\mathrm e}^{2 x -6} x^{2}}{2}\)	\(19\)
risch	\(x^{2}+\frac {x}{2}-\frac {{\mathrm e}^{2 x -6} x^{2}}{2}\)	\(19\)
default	\(x^{2}+\frac {x}{2}-\frac {3 \,{\mathrm e}^{2 x -6} \left (2 x -6\right )}{2}-\frac {9 \,{\mathrm e}^{2 x -6}}{2}-\frac {{\mathrm e}^{2 x -6} \left (2 x -6\right )^{2}}{8}\)	\(44\)
derivativedivides	\(\frac {13 x}{2}-\frac {39}{2}+\frac {\left (2 x -6\right )^{2}}{4}-\frac {3 \,{\mathrm e}^{2 x -6} \left (2 x -6\right )}{2}-\frac {9 \,{\mathrm e}^{2 x -6}}{2}-\frac {{\mathrm e}^{2 x -6} \left (2 x -6\right )^{2}}{8}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 18, normalized size = 0.82 \begin {gather*} -\frac {1}{2} \, x^{2} e^{\left (2 \, x - 6\right )} + x^{2} + \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^2*e^(2*x - 6) + x^2 + 1/2*x

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mupad [B]  time = 0.05, size = 18, normalized size = 0.82 \begin {gather*} \frac {x}{2}+x^2-\frac {x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-6}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/2 + x^2 - (x^2*exp(2*x)*exp(-6))/2

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sympy [A]  time = 0.09, size = 17, normalized size = 0.77 \begin {gather*} - \frac {x^{2} e^{2 x - 6}}{2} + x^{2} + \frac {x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2*exp(2*x - 6)/2 + x**2 + x/2

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