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

Optimal. Leaf size=18 \[ 8-2 x+\frac {e^x x}{3}-x^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {12, 2176, 2194} \begin {gather*} -x^2-2 x-\frac {e^x}{3}+\frac {1}{3} e^x (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 - 6*x + E^x*(1 + x))/3,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} \frac {1}{3} \left (e^x x-3 (1+x)^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 6*x + E^x*(1 + x))/3,x]

[Out]

(E^x*x - 3*(1 + x)^2)/3

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fricas [A]  time = 0.74, size = 14, normalized size = 0.78 \begin {gather*} -x^{2} + \frac {1}{3} \, x e^{x} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 + 1/3*x*e^x - 2*x

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giac [A]  time = 0.19, size = 14, normalized size = 0.78 \begin {gather*} -x^{2} + \frac {1}{3} \, x e^{x} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 + 1/3*x*e^x - 2*x

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maple [A]  time = 0.03, size = 15, normalized size = 0.83

method	result	size
default	\(-x^{2}-2 x +\frac {{\mathrm e}^{x} x}{3}\)	\(15\)
norman	\(-x^{2}-2 x +\frac {{\mathrm e}^{x} x}{3}\)	\(15\)
risch	\(-x^{2}-2 x +\frac {{\mathrm e}^{x} x}{3}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.38, size = 14, normalized size = 0.78 \begin {gather*} -x^{2} + \frac {1}{3} \, x e^{x} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 + 1/3*x*e^x - 2*x

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mupad [B]  time = 0.04, size = 12, normalized size = 0.67 \begin {gather*} -\frac {x\,\left (3\,x-{\mathrm {e}}^x+6\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(x + 1))/3 - 2*x - 2,x)

[Out]

-(x*(3*x - exp(x) + 6))/3

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sympy [A]  time = 0.08, size = 12, normalized size = 0.67 \begin {gather*} - x^{2} + \frac {x e^{x}}{3} - 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2 + x*exp(x)/3 - 2*x

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