3.60.9 \(\int e^{5-x} (-82+x) \, dx\)

Optimal. Leaf size=14 \[ 2-e^{5-x} (-81+x) \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.64, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2176, 2194} \begin {gather*} e^{5-x} (82-x)-e^{5-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(5 - x)*(-82 + x),x]

[Out]

-E^(5 - x) + E^(5 - x)*(82 - 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} &=e^{5-x} (82-x)+\int e^{5-x} \, dx\\ &=-e^{5-x}+e^{5-x} (82-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 12, normalized size = 0.86 \begin {gather*} -e^{5-x} (-81+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(5 - x)*(-82 + x),x]

[Out]

-(E^(5 - x)*(-81 + x))

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fricas [A]  time = 0.74, size = 11, normalized size = 0.79 \begin {gather*} -{\left (x - 81\right )} e^{\left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-82)/exp(x-5),x, algorithm="fricas")

[Out]

-(x - 81)*e^(-x + 5)

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giac [A]  time = 0.14, size = 11, normalized size = 0.79 \begin {gather*} -{\left (x - 81\right )} e^{\left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-82)/exp(x-5),x, algorithm="giac")

[Out]

-(x - 81)*e^(-x + 5)

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maple [A]  time = 0.07, size = 12, normalized size = 0.86

method	result	size
gosper	\(-{\mathrm e}^{5-x} \left (x -81\right )\)	\(12\)
norman	\(\left (81-x \right ) {\mathrm e}^{5-x}\)	\(13\)
risch	\(\left (81-x \right ) {\mathrm e}^{5-x}\)	\(13\)
derivativedivides	\(-{\mathrm e}^{5-x} \left (x -5\right )+76 \,{\mathrm e}^{5-x}\)	\(21\)
default	\(-{\mathrm e}^{5-x} \left (x -5\right )+76 \,{\mathrm e}^{5-x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-82)/exp(x-5),x,method=_RETURNVERBOSE)

[Out]

-1/exp(x-5)*(x-81)

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maxima [A]  time = 0.36, size = 22, normalized size = 1.57 \begin {gather*} -{\left (x e^{5} + e^{5}\right )} e^{\left (-x\right )} + 82 \, e^{\left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-82)/exp(x-5),x, algorithm="maxima")

[Out]

-(x*e^5 + e^5)*e^(-x) + 82*e^(-x + 5)

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mupad [B]  time = 0.05, size = 11, normalized size = 0.79 \begin {gather*} -{\mathrm {e}}^{5-x}\,\left (x-81\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(5 - x)*(x - 82),x)

[Out]

-exp(5 - x)*(x - 81)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.50 \begin {gather*} \left (81 - x\right ) e^{5 - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-82)/exp(x-5),x)

[Out]

(81 - x)*exp(5 - x)

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