3.194 \(\int e^{-0.1 x} x \, dx\)

Optimal. Leaf size=16 \[ -10. e^{-0.1 x} x-100. e^{-0.1 x} \]

[Out]

-100./E^(0.1*x) - (10.*x)/E^(0.1*x)

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Rubi [A]  time = 0.0075144, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2176, 2194} \[ -10. e^{-0.1 x} x-100. e^{-0.1 x} \]

Antiderivative was successfully verified.

[In]

Int[x/E^(0.1*x),x]

[Out]

-100./E^(0.1*x) - (10.*x)/E^(0.1*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{align*} \int e^{-0.1 x} x \, dx &=-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]  time = 0.0049634, size = 11, normalized size = 0.69 \[ e^{-0.1 x} (-10. x-100.) \]

Antiderivative was successfully verified.

[In]

Integrate[x/E^(0.1*x),x]

[Out]

(-99.99999999999999 - 10.*x)/E^(0.1*x)

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Maple [A]  time = 0.003, size = 10, normalized size = 0.6 \begin{align*} - 10.0\, \left ( x+ 10.0 \right ){{\rm e}^{- 0.1000000000\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-.1*x)*x,x)

[Out]

-10.*(x+10.)*exp(-.1000000000*x)

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Maxima [A]  time = 1.04478, size = 12, normalized size = 0.75 \begin{align*} -10 \,{\left (x + 10\right )} e^{\left (-\frac{1}{10} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*(x + 10)*e^(-1/10*x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0.090221, size = 10, normalized size = 0.62 \begin{align*} 1.0 \left (- 10.0 x - 100.0\right ) e^{- 0.1 x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-.1*x)*x,x)

[Out]

1.0*(-10.0*x - 100.0)*exp(-0.1*x)

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Giac [A]  time = 1.21675, size = 14, normalized size = 0.88 \begin{align*}{\left (-10.0 \, x - 100.0\right )} e^{\left (-0.1 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-10.0*x - 100.0)*e^(-0.1*x)