\(\int e^{-x/10} x \, dx\) [194]

Optimal result

Integrand size = 9, antiderivative size = 20 \[ \int e^{-x/10} x \, dx=-100 e^{-x/10}-10 e^{-x/10} x \]

[Out]

-100*exp(-1/10*x)-10*exp(-1/10*x)*x

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2207, 2225} \[ \int e^{-x/10} x \, dx=-10 e^{-x/10} x-100 e^{-x/10} \]

[In]

Int[x/E^(x/10),x]

[Out]

-100/E^(x/10) - (10*x)/E^(x/10)

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}& = -10. e^{-0.1 x} x+10. \int e^{-0.1 x} \, dx \\ & = -100. e^{-0.1 x}-10. e^{-0.1 x} x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.65 \[ \int e^{-x/10} x \, dx=e^{-x/10} (-100-10 x) \]

[In]

Integrate[x/E^(x/10),x]

[Out]

(-100 - 10*x)/E^(x/10)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.50

method	result	size
gosper	\(-10 \left (x +10\right ) {\mathrm e}^{-\frac {x}{10}}\)	\(10\)
risch	\(\left (-100-10 x \right ) {\mathrm e}^{-\frac {x}{10}}\)	\(11\)
meijerg	\(100-50 \left (2+\frac {x}{5}\right ) {\mathrm e}^{-\frac {x}{10}}\)	\(14\)
derivativedivides	\(-100 \,{\mathrm e}^{-\frac {x}{10}}-10 \,{\mathrm e}^{-\frac {x}{10}} x\)	\(15\)
default	\(-100 \,{\mathrm e}^{-\frac {x}{10}}-10 \,{\mathrm e}^{-\frac {x}{10}} x\)	\(15\)
norman	\(-100 \,{\mathrm e}^{-\frac {x}{10}}-10 \,{\mathrm e}^{-\frac {x}{10}} x\)	\(15\)
parallelrisch	\(-100 \,{\mathrm e}^{-\frac {x}{10}}-10 \,{\mathrm e}^{-\frac {x}{10}} x\)	\(15\)
parts	\(-100 \,{\mathrm e}^{-\frac {x}{10}}-10 \,{\mathrm e}^{-\frac {x}{10}} x\)	\(15\)

[In]

int(exp(-1/10*x)*x,x,method=_RETURNVERBOSE)

[Out]

-10*(x+10)*exp(-1/10*x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.45 \[ \int e^{-x/10} x \, dx=-10 \, {\left (x + 10\right )} e^{\left (-\frac {1}{10} \, x\right )} \]

[In]

integrate(exp(-1/10*x)*x,x, algorithm="fricas")

[Out]

-10*(x + 10)*e^(-1/10*x)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.50 \[ \int e^{-x/10} x \, dx=\left (- 10 x - 100\right ) e^{- \frac {x}{10}} \]

[In]

integrate(exp(-1/10*x)*x,x)

[Out]

(-10*x - 100)*exp(-x/10)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.45 \[ \int e^{-x/10} x \, dx=-10 \, {\left (x + 10\right )} e^{\left (-\frac {1}{10} \, x\right )} \]

[In]

integrate(exp(-1/10*x)*x,x, algorithm="maxima")

[Out]

-10*(x + 10)*e^(-1/10*x)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.45 \[ \int e^{-x/10} x \, dx=-10 \, {\left (x + 10\right )} e^{\left (-\frac {1}{10} \, x\right )} \]

[In]

integrate(exp(-1/10*x)*x,x, algorithm="giac")

[Out]

-10*(x + 10)*e^(-1/10*x)

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.45 \[ \int e^{-x/10} x \, dx=-10\,{\mathrm {e}}^{-\frac {x}{10}}\,\left (x+10\right ) \]

[In]

int(x*exp(-x/10),x)

[Out]

-10*exp(-x/10)*(x + 10)