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.*exp(-.1*x)-10.*exp(-.1*x)*x

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred when FriCAS evaluated 'integrate(sage0,sage1)':   There are 1
2 exposed and 10 unexposed library operations named       integrate having 2 argument(s) but none was determin
ed to be       applicable. Use HyperDoc Browse, or issue                             )display op integrate
to learn more about the available operations. Perhaps       package-calling the operation or using coercions
on the arguments       will allow you to apply the operation.    Cannot find a definition or applicable libra
ry operation named       integrate with argument type(s)                                Expression(Float)
Variable(x)            Perhaps you should use @ to indicate the required return t
ype, or       $ to specify which version of the function you need.

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giac [A]  time = 0.21, size = 10, normalized size = 0.62 \[ {\left (-10.0000000000000 \, x - 100.000000000000\right )} e^{\left (-0.100000000000000 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-10.0000000000000*x - 100.000000000000)*e^(-0.100000000000000*x)

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maple [A]  time = 0.00, size = 10, normalized size = 0.62 \[ - 10.0 \left (x + 10.0\right ) {\mathrm e}^{- 0.1000000000 x} \]

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 = 0.52, size = 9, normalized size = 0.56 \[ -10 \, {\left (x + 10\right )} e^{\left (-\frac {1}{10} \, x\right )} \]

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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mupad [B]  time = 0.03, size = 9, normalized size = 0.56 \[ -10\,{\mathrm {e}}^{-0.1\,x}\,\left (x+10.0\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(-0.1*x),x)

[Out]

-10*exp(-0.1*x)*(x + 10.0)

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sympy [A]  time = 0.09, size = 10, normalized size = 0.62 \[ 1.0 \left (- 10.0 x - 100.0\right ) e^{- 0.1 x} \]

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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