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

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

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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} $$\begin{array}{c}-e^{2x-6}{x}^8\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int e^{-6+2x}(-8-2x)x^{7}dx\\ & =\int (-8e^{-6+2x}{x}^7-2e^{-6+2x}{x}^8)dx\\ & =-(2\int e^{-6+2x}{x}^{8}dx)-8\int e^{-6+2x}{x}^{7}dx\\ & =-4e^{-6+2x}{x}^7-e^{-6+2x}{x}^8+8\int e^{-6+2x}{x}^{7}dx+28\int e^{-6+2x}{x}^{6}dx\\ & =14e^{-6+2x}{x}^6-e^{-6+2x}{x}^8-28\int e^{-6+2x}{x}^{6}dx-84\int e^{-6+2x}{x}^{5}dx\\ & =-42e^{-6+2x}{x}^5-e^{-6+2x}{x}^8+84\int e^{-6+2x}{x}^{5}dx+210\int e^{-6+2x}{x}^{4}dx\\ & =105e^{-6+2x}{x}^4-e^{-6+2x}{x}^8-210\int e^{-6+2x}{x}^{4}dx-420\int e^{-6+2x}{x}^{3}dx\\ & =-210e^{-6+2x}{x}^3-e^{-6+2x}{x}^8+420\int e^{-6+2x}{x}^{3}dx+630\int e^{-6+2x}{x}^{2}dx\\ & =315e^{-6+2x}{x}^2-e^{-6+2x}{x}^8-630\int e^{-6+2x}xdx-630\int e^{-6+2x}{x}^{2}dx\\ & =-315e^{-6+2x}x-e^{-6+2x}{x}^8+315\int e^{-6+2x}dx+630\int e^{-6+2x}xdx\\ & =\frac{315}{2}{e}^{-6+2x}-e^{-6+2x}{x}^8-315\int e^{-6+2x}dx\\ & =-e^{-6+2x}{x}^8\end{array}\end{array}$$

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

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 $$\begin{array}{c}-x^8{e}^{(2x-6)}\end{array}$$

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 $$\begin{array}{c}-x^8{e}^{(2x-6)}\end{array}$$

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

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 $$\begin{array}{c}-\frac{1}{2}(2x^8-8x^7+28x^6-84x^5+210x^4-420x^3+630x^2-630x+315)e^{(2x-6)}-\frac{1}{2}(8x^7-28x^6+84x^5-210x^4+420x^3-630x^2+630x-315)e^{(2x-6)}\end{array}$$

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 $$\begin{array}{c}-x^8{\mathrm{e}}^{2x}{\mathrm{e}}^{-6}\end{array}$$

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 $$\begin{array}{c}-x^8{e}^{2x-6}\end{array}$$

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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