3.17.71 \(\int e^{-6+2 x} (-8 x^7-2 x^8) \, dx\)

Optimal. Leaf size=16 \[ \frac {5}{2}-e^{-6+2 x} x^8 \]

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Rubi [A]  time = 0.36, antiderivative size = 12, normalized size of antiderivative = 0.75, number of steps used = 20, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1593, 2196, 2176, 2194} \begin {gather*} -e^{2 x-6} x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-6 + 2*x)*(-8*x^7 - 2*x^8),x]

[Out]

-(E^(-6 + 2*x)*x^8)

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-6+2 x} (-8-2 x) x^7 \, dx\\ &=\int \left (-8 e^{-6+2 x} x^7-2 e^{-6+2 x} x^8\right ) \, dx\\ &=-\left (2 \int e^{-6+2 x} x^8 \, dx\right )-8 \int e^{-6+2 x} x^7 \, dx\\ &=-4 e^{-6+2 x} x^7-e^{-6+2 x} x^8+8 \int e^{-6+2 x} x^7 \, dx+28 \int e^{-6+2 x} x^6 \, dx\\ &=14 e^{-6+2 x} x^6-e^{-6+2 x} x^8-28 \int e^{-6+2 x} x^6 \, dx-84 \int e^{-6+2 x} x^5 \, dx\\ &=-42 e^{-6+2 x} x^5-e^{-6+2 x} x^8+84 \int e^{-6+2 x} x^5 \, dx+210 \int e^{-6+2 x} x^4 \, dx\\ &=105 e^{-6+2 x} x^4-e^{-6+2 x} x^8-210 \int e^{-6+2 x} x^4 \, dx-420 \int e^{-6+2 x} x^3 \, dx\\ &=-210 e^{-6+2 x} x^3-e^{-6+2 x} x^8+420 \int e^{-6+2 x} x^3 \, dx+630 \int e^{-6+2 x} x^2 \, dx\\ &=315 e^{-6+2 x} x^2-e^{-6+2 x} x^8-630 \int e^{-6+2 x} x \, dx-630 \int e^{-6+2 x} x^2 \, dx\\ &=-315 e^{-6+2 x} x-e^{-6+2 x} x^8+315 \int e^{-6+2 x} \, dx+630 \int e^{-6+2 x} x \, dx\\ &=\frac {315}{2} e^{-6+2 x}-e^{-6+2 x} x^8-315 \int e^{-6+2 x} \, dx\\ &=-e^{-6+2 x} x^8\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 0.75 \begin {gather*} -e^{-6+2 x} x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-6 + 2*x)*(-8*x^7 - 2*x^8),x]

[Out]

-(E^(-6 + 2*x)*x^8)

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fricas [A]  time = 0.65, size = 11, normalized size = 0.69 \begin {gather*} -x^{8} e^{\left (2 \, x - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^8-8*x^7)/exp(3-x)^2,x, algorithm="fricas")

[Out]

-x^8*e^(2*x - 6)

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giac [A]  time = 0.21, size = 11, normalized size = 0.69 \begin {gather*} -x^{8} e^{\left (2 \, x - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^8-8*x^7)/exp(3-x)^2,x, algorithm="giac")

[Out]

-x^8*e^(2*x - 6)

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maple [A]  time = 0.07, size = 12, normalized size = 0.75

method	result	size
risch	\(-x^{8} {\mathrm e}^{2 x -6}\)	\(12\)
gosper	\(-x^{8} {\mathrm e}^{2 x -6}\)	\(14\)
norman	\(-x^{8} {\mathrm e}^{2 x -6}\)	\(14\)
derivativedivides	\(-6561 \,{\mathrm e}^{2 x -6}+17496 \,{\mathrm e}^{2 x -6} \left (3-x \right )-20412 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{2}+13608 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{3}-5670 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{4}+1512 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{5}-252 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{6}+24 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{7}-{\mathrm e}^{2 x -6} \left (3-x \right )^{8}\)	\(146\)
default	\(-6561 \,{\mathrm e}^{2 x -6}+17496 \,{\mathrm e}^{2 x -6} \left (3-x \right )-20412 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{2}+13608 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{3}-5670 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{4}+1512 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{5}-252 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{6}+24 \,{\mathrm e}^{2 x -6} \left (3-x \right )^{7}-{\mathrm e}^{2 x -6} \left (3-x \right )^{8}\)	\(146\)
meijerg	\(\frac {{\mathrm e}^{48+2 x -2 x \,{\mathrm e}^{-6}} \left (40320-\frac {\left (2304 x^{8} {\mathrm e}^{-48}-9216 x^{7} {\mathrm e}^{-42}+32256 x^{6} {\mathrm e}^{-36}-96768 x^{5} {\mathrm e}^{-30}+241920 x^{4} {\mathrm e}^{-24}-483840 x^{3} {\mathrm e}^{-18}+725760 x^{2} {\mathrm e}^{-12}-725760 x \,{\mathrm e}^{-6}+362880\right ) {\mathrm e}^{2 x \,{\mathrm e}^{-6}}}{9}\right )}{256}-\frac {{\mathrm e}^{2 x +42-2 x \,{\mathrm e}^{-6}} \left (5040-\frac {\left (-1024 x^{7} {\mathrm e}^{-42}+3584 x^{6} {\mathrm e}^{-36}-10752 x^{5} {\mathrm e}^{-30}+26880 x^{4} {\mathrm e}^{-24}-53760 x^{3} {\mathrm e}^{-18}+80640 x^{2} {\mathrm e}^{-12}-80640 x \,{\mathrm e}^{-6}+40320\right ) {\mathrm e}^{2 x \,{\mathrm e}^{-6}}}{8}\right )}{32}\)	\(153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^8-8*x^7)/exp(3-x)^2,x,method=_RETURNVERBOSE)

[Out]

-x^8*exp(2*x-6)

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maxima [B]  time = 0.39, size = 92, normalized size = 5.75 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x^{8} - 8 \, x^{7} + 28 \, x^{6} - 84 \, x^{5} + 210 \, x^{4} - 420 \, x^{3} + 630 \, x^{2} - 630 \, x + 315\right )} e^{\left (2 \, x - 6\right )} - \frac {1}{2} \, {\left (8 \, x^{7} - 28 \, x^{6} + 84 \, x^{5} - 210 \, x^{4} + 420 \, x^{3} - 630 \, x^{2} + 630 \, x - 315\right )} e^{\left (2 \, x - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^8-8*x^7)/exp(3-x)^2,x, algorithm="maxima")

[Out]

-1/2*(2*x^8 - 8*x^7 + 28*x^6 - 84*x^5 + 210*x^4 - 420*x^3 + 630*x^2 - 630*x + 315)*e^(2*x - 6) - 1/2*(8*x^7 -
28*x^6 + 84*x^5 - 210*x^4 + 420*x^3 - 630*x^2 + 630*x - 315)*e^(2*x - 6)

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mupad [B]  time = 1.12, size = 11, normalized size = 0.69 \begin {gather*} -x^8\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(2*x - 6)*(8*x^7 + 2*x^8),x)

[Out]

-x^8*exp(2*x)*exp(-6)

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sympy [A]  time = 0.10, size = 10, normalized size = 0.62 \begin {gather*} - x^{8} e^{2 x - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**8-8*x**7)/exp(3-x)**2,x)

[Out]

-x**8*exp(2*x - 6)

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