3.6.27 \(\int (8752+e^x (-4-4 x)) \, dx\)

Optimal. Leaf size=12 \[ 4 \left (2188 x-e^x x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.42, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2176, 2194} \begin {gather*} 8752 x+4 e^x-4 e^x (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[8752 + E^x*(-4 - 4*x),x]

[Out]

4*E^x + 8752*x - 4*E^x*(1 + 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} &=8752 x+\int e^x (-4-4 x) \, dx\\ &=8752 x-4 e^x (1+x)+4 \int e^x \, dx\\ &=4 e^x+8752 x-4 e^x (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.83 \begin {gather*} 8752 x-4 e^x x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[8752 + E^x*(-4 - 4*x),x]

[Out]

8752*x - 4*E^x*x

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fricas [A]  time = 0.70, size = 9, normalized size = 0.75 \begin {gather*} -4 \, x e^{x} + 8752 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-4)*exp(x)+8752,x, algorithm="fricas")

[Out]

-4*x*e^x + 8752*x

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giac [A]  time = 0.34, size = 9, normalized size = 0.75 \begin {gather*} -4 \, x e^{x} + 8752 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-4)*exp(x)+8752,x, algorithm="giac")

[Out]

-4*x*e^x + 8752*x

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maple [A]  time = 0.01, size = 10, normalized size = 0.83

method	result	size
default	\(8752 x -4 \,{\mathrm e}^{x} x\)	\(10\)
norman	\(8752 x -4 \,{\mathrm e}^{x} x\)	\(10\)
risch	\(8752 x -4 \,{\mathrm e}^{x} x\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x-4)*exp(x)+8752,x,method=_RETURNVERBOSE)

[Out]

8752*x-4*exp(x)*x

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maxima [A]  time = 0.37, size = 15, normalized size = 1.25 \begin {gather*} -4 \, {\left (x - 1\right )} e^{x} + 8752 \, x - 4 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-4)*exp(x)+8752,x, algorithm="maxima")

[Out]

-4*(x - 1)*e^x + 8752*x - 4*e^x

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mupad [B]  time = 0.04, size = 7, normalized size = 0.58 \begin {gather*} -4\,x\,\left ({\mathrm {e}}^x-2188\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8752 - exp(x)*(4*x + 4),x)

[Out]

-4*x*(exp(x) - 2188)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.67 \begin {gather*} - 4 x e^{x} + 8752 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-4)*exp(x)+8752,x)

[Out]

-4*x*exp(x) + 8752*x

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