3.4.14 \(\int \frac {1}{8} e^{-x} (16+e^x (-21-16 x)-16 x) \, dx\)

Optimal. Leaf size=29 \[ 2 \left (6-\frac {13 x}{16}+e^{-x} x+\frac {1}{2} \left (-x-x^2\right )\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 19, normalized size of antiderivative = 0.66, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {12, 6742, 2194, 2176} \begin {gather*} -x^2+2 e^{-x} x-\frac {21 x}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(16 + E^x*(-21 - 16*x) - 16*x)/(8*E^x),x]

[Out]

(-21*x)/8 + (2*x)/E^x - x^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int e^{-x} \left (16+e^x (-21-16 x)-16 x\right ) \, dx\\ &=\frac {1}{8} \int \left (-21+16 e^{-x}-16 x-16 e^{-x} x\right ) \, dx\\ &=-\frac {21 x}{8}-x^2+2 \int e^{-x} \, dx-2 \int e^{-x} x \, dx\\ &=-2 e^{-x}-\frac {21 x}{8}+2 e^{-x} x-x^2-2 \int e^{-x} \, dx\\ &=-\frac {21 x}{8}+2 e^{-x} x-x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 0.66 \begin {gather*} -\frac {21 x}{8}+2 e^{-x} x-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16 + E^x*(-21 - 16*x) - 16*x)/(8*E^x),x]

[Out]

(-21*x)/8 + (2*x)/E^x - x^2

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fricas [A]  time = 0.72, size = 22, normalized size = 0.76 \begin {gather*} -\frac {1}{8} \, {\left ({\left (8 \, x^{2} + 21 \, x\right )} e^{x} - 16 \, x\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-16*x-21)*exp(x)-16*x+16)/exp(x),x, algorithm="fricas")

[Out]

-1/8*((8*x^2 + 21*x)*e^x - 16*x)*e^(-x)

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giac [A]  time = 0.24, size = 16, normalized size = 0.55 \begin {gather*} -x^{2} + 2 \, x e^{\left (-x\right )} - \frac {21}{8} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-16*x-21)*exp(x)-16*x+16)/exp(x),x, algorithm="giac")

[Out]

-x^2 + 2*x*e^(-x) - 21/8*x

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maple [A]  time = 0.02, size = 17, normalized size = 0.59

method	result	size
default	\(-x^{2}-\frac {21 x}{8}+2 x \,{\mathrm e}^{-x}\)	\(17\)
risch	\(-x^{2}-\frac {21 x}{8}+2 x \,{\mathrm e}^{-x}\)	\(17\)
norman	\(\left (2 x -\frac {21 \,{\mathrm e}^{x} x}{8}-{\mathrm e}^{x} x^{2}\right ) {\mathrm e}^{-x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*((-16*x-21)*exp(x)-16*x+16)/exp(x),x,method=_RETURNVERBOSE)

[Out]

-x^2-21/8*x+2*x/exp(x)

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maxima [A]  time = 0.37, size = 24, normalized size = 0.83 \begin {gather*} -x^{2} + 2 \, {\left (x + 1\right )} e^{\left (-x\right )} - \frac {21}{8} \, x - 2 \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-16*x-21)*exp(x)-16*x+16)/exp(x),x, algorithm="maxima")

[Out]

-x^2 + 2*(x + 1)*e^(-x) - 21/8*x - 2*e^(-x)

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mupad [B]  time = 0.34, size = 14, normalized size = 0.48 \begin {gather*} -\frac {x\,\left (8\,x-16\,{\mathrm {e}}^{-x}+21\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*(2*x + (exp(x)*(16*x + 21))/8 - 2),x)

[Out]

-(x*(8*x - 16*exp(-x) + 21))/8

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sympy [A]  time = 0.09, size = 14, normalized size = 0.48 \begin {gather*} - x^{2} - \frac {21 x}{8} + 2 x e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-16*x-21)*exp(x)-16*x+16)/exp(x),x)

[Out]

-x**2 - 21*x/8 + 2*x*exp(-x)

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