### 3.39 $$\int e^{-x} x^2 \, dx$$

Optimal. Leaf size=26 $-e^{-x} x^2-2 e^{-x} x-2 e^{-x}$

[Out]

-2/E^x - (2*x)/E^x - x^2/E^x

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Rubi [A]  time = 0.0179771, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {2176, 2194} $-e^{-x} x^2-2 e^{-x} x-2 e^{-x}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2/E^x,x]

[Out]

-2/E^x - (2*x)/E^x - x^2/E^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^{-x} x^2 \, dx &=-e^{-x} x^2+2 \int e^{-x} x \, dx\\ &=-2 e^{-x} x-e^{-x} x^2+2 \int e^{-x} \, dx\\ &=-2 e^{-x}-2 e^{-x} x-e^{-x} x^2\\ \end{align*}

Mathematica [A]  time = 0.0059813, size = 16, normalized size = 0.62 $e^{-x} \left (-x^2-2 x-2\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2/E^x,x]

[Out]

(-2 - 2*x - x^2)/E^x

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Maple [A]  time = 0.001, size = 15, normalized size = 0.6 \begin{align*} -{\frac{{x}^{2}+2\,x+2}{{{\rm e}^{x}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/exp(x),x)

[Out]

-(x^2+2*x+2)/exp(x)

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Maxima [A]  time = 0.938155, size = 19, normalized size = 0.73 \begin{align*} -{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/exp(x),x, algorithm="maxima")

[Out]

-(x^2 + 2*x + 2)*e^(-x)

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Fricas [A]  time = 2.06137, size = 34, normalized size = 1.31 \begin{align*} -{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/exp(x),x, algorithm="fricas")

[Out]

-(x^2 + 2*x + 2)*e^(-x)

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Sympy [A]  time = 0.078733, size = 12, normalized size = 0.46 \begin{align*} \left (- x^{2} - 2 x - 2\right ) e^{- x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/exp(x),x)

[Out]

(-x**2 - 2*x - 2)*exp(-x)

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Giac [A]  time = 1.05395, size = 19, normalized size = 0.73 \begin{align*} -{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/exp(x),x, algorithm="giac")

[Out]

-(x^2 + 2*x + 2)*e^(-x)