\(\int e^{5-x} (-82+x) \, dx\) [5909]

Optimal result

Integrand size = 11, antiderivative size = 14 \[ \int e^{5-x} (-82+x) \, dx=2-e^{5-x} (-81+x) \]

[Out]

2-1/exp(-5+x)*(x-81)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2207, 2225} \[ \int e^{5-x} (-82+x) \, dx=e^{5-x} (82-x)-e^{5-x} \]

[In]

Int[E^(5 - x)*(-82 + x),x]

[Out]

-E^(5 - x) + E^(5 - x)*(82 - x)

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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{align*} \text {integral}& = e^{5-x} (82-x)+\int e^{5-x} \, dx \\ & = -e^{5-x}+e^{5-x} (82-x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int e^{5-x} (-82+x) \, dx=-e^{5-x} (-81+x) \]

[In]

Integrate[E^(5 - x)*(-82 + x),x]

[Out]

-(E^(5 - x)*(-81 + x))

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 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	\(\left (81-x \right ) {\mathrm e}^{5-x}\)	\(13\)
parallelrisch	\(\left (81-x \right ) {\mathrm e}^{5-x}\)	\(13\)
derivativedivides	\(-{\mathrm e}^{5-x} \left (-5+x \right )+76 \,{\mathrm e}^{5-x}\)	\(21\)
default	\(-{\mathrm e}^{5-x} \left (-5+x \right )+76 \,{\mathrm e}^{5-x}\)	\(21\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int e^{5-x} (-82+x) \, dx=-{\left (x - 81\right )} e^{\left (-x + 5\right )} \]

[In]

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

[Out]

-(x - 81)*e^(-x + 5)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.50 \[ \int e^{5-x} (-82+x) \, dx=\left (81 - x\right ) e^{5 - x} \]

[In]

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

[Out]

(81 - x)*exp(5 - x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int e^{5-x} (-82+x) \, dx=-{\left (x e^{5} + e^{5}\right )} e^{\left (-x\right )} + 82 \, e^{\left (-x + 5\right )} \]

[In]

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

[Out]

-(x*e^5 + e^5)*e^(-x) + 82*e^(-x + 5)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int e^{5-x} (-82+x) \, dx=-{\left (x - 81\right )} e^{\left (-x + 5\right )} \]

[In]

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

[Out]

-(x - 81)*e^(-x + 5)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int e^{5-x} (-82+x) \, dx=-{\mathrm {e}}^{5-x}\,\left (x-81\right ) \]

[In]

int(exp(5 - x)*(x - 82),x)

[Out]

-exp(5 - x)*(x - 81)