3.41.67 \(\int \frac {1}{9} e^{-24-x} (27 e^{24+4 x}+e^{24} (-9+9 x)+e^x (32 x^7-256 e^3 x^7+896 e^6 x^7-1792 e^9 x^7+2240 e^{12} x^7-1792 e^{15} x^7+896 e^{18} x^7-256 e^{21} x^7+32 e^{24} x^7)) \, dx\)

Optimal. Leaf size=31 \[ -e^{-x} \left (-e^{4 x}+x\right )+\frac {4}{9} \left (x-\frac {x}{e^3}\right )^8 \]

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 43, normalized size of antiderivative = 1.39, number of steps used = 6, number of rules used = 4, integrand size = 104, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 6742, 2194, 2176} \begin {gather*} \frac {4 \left (1-e^3\right )^8 x^8}{9 e^{24}}-e^{-x}+e^{3 x}+e^{-x} (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-24 - x)*(27*E^(24 + 4*x) + E^24*(-9 + 9*x) + E^x*(32*x^7 - 256*E^3*x^7 + 896*E^6*x^7 - 1792*E^9*x^7 +
2240*E^12*x^7 - 1792*E^15*x^7 + 896*E^18*x^7 - 256*E^21*x^7 + 32*E^24*x^7)))/9,x]

[Out]

-E^(-x) + E^(3*x) + (1 - x)/E^x + (4*(1 - E^3)^8*x^8)/(9*E^24)

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}{9} \int e^{-24-x} \left (27 e^{24+4 x}+e^{24} (-9+9 x)+e^x \left (32 x^7-256 e^3 x^7+896 e^6 x^7-1792 e^9 x^7+2240 e^{12} x^7-1792 e^{15} x^7+896 e^{18} x^7-256 e^{21} x^7+32 e^{24} x^7\right )\right ) \, dx\\ &=\frac {1}{9} \int \left (27 e^{3 x}+9 e^{-x} (-1+x)+\frac {32 \left (-1+e^3\right )^8 x^7}{e^{24}}\right ) \, dx\\ &=\frac {4 \left (1-e^3\right )^8 x^8}{9 e^{24}}+3 \int e^{3 x} \, dx+\int e^{-x} (-1+x) \, dx\\ &=e^{3 x}+e^{-x} (1-x)+\frac {4 \left (1-e^3\right )^8 x^8}{9 e^{24}}+\int e^{-x} \, dx\\ &=-e^{-x}+e^{3 x}+e^{-x} (1-x)+\frac {4 \left (1-e^3\right )^8 x^8}{9 e^{24}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 31, normalized size = 1.00 \begin {gather*} e^{3 x}-e^{-x} x+\frac {4 \left (-1+e^3\right )^8 x^8}{9 e^{24}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-24 - x)*(27*E^(24 + 4*x) + E^24*(-9 + 9*x) + E^x*(32*x^7 - 256*E^3*x^7 + 896*E^6*x^7 - 1792*E^9
*x^7 + 2240*E^12*x^7 - 1792*E^15*x^7 + 896*E^18*x^7 - 256*E^21*x^7 + 32*E^24*x^7)))/9,x]

[Out]

E^(3*x) - x/E^x + (4*(-1 + E^3)^8*x^8)/(9*E^24)

________________________________________________________________________________________

fricas [B]  time = 1.37, size = 85, normalized size = 2.74 \begin {gather*} -\frac {1}{9} \, {\left (9 \, x e^{24} - 4 \, {\left (x^{8} e^{24} - 8 \, x^{8} e^{21} + 28 \, x^{8} e^{18} - 56 \, x^{8} e^{15} + 70 \, x^{8} e^{12} - 56 \, x^{8} e^{9} + 28 \, x^{8} e^{6} - 8 \, x^{8} e^{3} + x^{8}\right )} e^{x} - 9 \, e^{\left (4 \, x + 24\right )}\right )} e^{\left (-x - 24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896*x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*
x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7*exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x
),x, algorithm="fricas")

[Out]

-1/9*(9*x*e^24 - 4*(x^8*e^24 - 8*x^8*e^21 + 28*x^8*e^18 - 56*x^8*e^15 + 70*x^8*e^12 - 56*x^8*e^9 + 28*x^8*e^6
- 8*x^8*e^3 + x^8)*e^x - 9*e^(4*x + 24))*e^(-x - 24)

________________________________________________________________________________________

giac [B]  time = 0.16, size = 85, normalized size = 2.74 \begin {gather*} \frac {1}{9} \, {\left (4 \, x^{8} e^{108} - 32 \, x^{8} e^{105} + 112 \, x^{8} e^{102} - 224 \, x^{8} e^{99} + 280 \, x^{8} e^{96} - 224 \, x^{8} e^{93} + 112 \, x^{8} e^{90} - 32 \, x^{8} e^{87} + 4 \, x^{8} e^{84} - 9 \, x e^{\left (-x + 108\right )} + 9 \, e^{\left (3 \, x + 108\right )}\right )} e^{\left (-108\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896*x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*
x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7*exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x
),x, algorithm="giac")

[Out]

1/9*(4*x^8*e^108 - 32*x^8*e^105 + 112*x^8*e^102 - 224*x^8*e^99 + 280*x^8*e^96 - 224*x^8*e^93 + 112*x^8*e^90 -
32*x^8*e^87 + 4*x^8*e^84 - 9*x*e^(-x + 108) + 9*e^(3*x + 108))*e^(-108)

________________________________________________________________________________________

maple [B]  time = 0.03, size = 92, normalized size = 2.97

method	result	size
risch	\(-\frac {32 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{3}}{9}+\frac {4 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{24}}{9}-\frac {32 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{21}}{9}+\frac {112 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{18}}{9}-\frac {224 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{15}}{9}+\frac {280 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{12}}{9}-\frac {224 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{9}}{9}+\frac {112 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{6}}{9}+\frac {4 \,{\mathrm e}^{-24} x^{8}}{9}+{\mathrm e}^{3 x}-x \,{\mathrm e}^{-x}\)	\(92\)
default	\(\frac {{\mathrm e}^{-24} \left (4 x^{8}-32 x^{8} {\mathrm e}^{3}+112 x^{8} {\mathrm e}^{6}-224 x^{8} {\mathrm e}^{9}+280 x^{8} {\mathrm e}^{12}-224 x^{8} {\mathrm e}^{15}+112 x^{8} {\mathrm e}^{18}-32 x^{8} {\mathrm e}^{21}+4 x^{8} {\mathrm e}^{24}+9 \,{\mathrm e}^{-x} {\mathrm e}^{24}+9 \,{\mathrm e}^{3 x} {\mathrm e}^{24}+9 \,{\mathrm e}^{24} \left (-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}\right )\right )}{9}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896*x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*x^7*ex
p(3)^4-1792*x^7*exp(3)^3+896*x^7*exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x),x,me
thod=_RETURNVERBOSE)

[Out]

-32/9*exp(-24)*x^8*exp(3)+4/9*exp(-24)*x^8*exp(24)-32/9*exp(-24)*x^8*exp(21)+112/9*exp(-24)*x^8*exp(18)-224/9*
exp(-24)*x^8*exp(15)+280/9*exp(-24)*x^8*exp(12)-224/9*exp(-24)*x^8*exp(9)+112/9*exp(-24)*x^8*exp(6)+4/9*exp(-2
4)*x^8+exp(3*x)-x*exp(-x)

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 79, normalized size = 2.55 \begin {gather*} -\frac {32}{9} \, x^{8} e^{\left (-3\right )} + \frac {112}{9} \, x^{8} e^{\left (-6\right )} - \frac {224}{9} \, x^{8} e^{\left (-9\right )} + \frac {280}{9} \, x^{8} e^{\left (-12\right )} - \frac {224}{9} \, x^{8} e^{\left (-15\right )} + \frac {112}{9} \, x^{8} e^{\left (-18\right )} - \frac {32}{9} \, x^{8} e^{\left (-21\right )} + \frac {4}{9} \, x^{8} e^{\left (-24\right )} + \frac {4}{9} \, x^{8} - {\left (x + 1\right )} e^{\left (-x\right )} + e^{\left (3 \, x\right )} + e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896*x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*
x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7*exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x
),x, algorithm="maxima")

[Out]

-32/9*x^8*e^(-3) + 112/9*x^8*e^(-6) - 224/9*x^8*e^(-9) + 280/9*x^8*e^(-12) - 224/9*x^8*e^(-15) + 112/9*x^8*e^(
-18) - 32/9*x^8*e^(-21) + 4/9*x^8*e^(-24) + 4/9*x^8 - (x + 1)*e^(-x) + e^(3*x) + e^(-x)

________________________________________________________________________________________

mupad [B]  time = 3.15, size = 50, normalized size = 1.61 \begin {gather*} {\mathrm {e}}^{3\,x}-x\,{\mathrm {e}}^{-x}+x^8\,\left (\frac {112\,{\mathrm {e}}^{-6}}{9}-\frac {32\,{\mathrm {e}}^{-3}}{9}-\frac {224\,{\mathrm {e}}^{-9}}{9}+\frac {280\,{\mathrm {e}}^{-12}}{9}-\frac {224\,{\mathrm {e}}^{-15}}{9}+\frac {112\,{\mathrm {e}}^{-18}}{9}-\frac {32\,{\mathrm {e}}^{-21}}{9}+\frac {4\,{\mathrm {e}}^{-24}}{9}+\frac {4}{9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*exp(-24)*(3*exp(4*x)*exp(24) + (exp(x)*(896*x^7*exp(6) - 256*x^7*exp(3) - 1792*x^7*exp(9) + 2240*x
^7*exp(12) - 1792*x^7*exp(15) + 896*x^7*exp(18) - 256*x^7*exp(21) + 32*x^7*exp(24) + 32*x^7))/9 + (exp(24)*(9*
x - 9))/9),x)

[Out]

exp(3*x) - x*exp(-x) + x^8*((112*exp(-6))/9 - (32*exp(-3))/9 - (224*exp(-9))/9 + (280*exp(-12))/9 - (224*exp(-
15))/9 + (112*exp(-18))/9 - (32*exp(-21))/9 + (4*exp(-24))/9 + 4/9)

________________________________________________________________________________________

sympy [B]  time = 0.23, size = 60, normalized size = 1.94 \begin {gather*} \frac {x^{8} \left (- 32 e^{21} - 224 e^{15} - 224 e^{9} - 32 e^{3} + 4 + 112 e^{6} + 280 e^{12} + 112 e^{18} + 4 e^{24}\right )}{9 e^{24}} - x e^{- x} + e^{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(27*exp(3)**8*exp(4*x)+(32*x**7*exp(3)**8-256*x**7*exp(3)**7+896*x**7*exp(3)**6-1792*x**7*exp(3)
**5+2240*x**7*exp(3)**4-1792*x**7*exp(3)**3+896*x**7*exp(3)**2-256*x**7*exp(3)+32*x**7)*exp(x)+(9*x-9)*exp(3)*
*8)/exp(3)**8/exp(x),x)

[Out]

x**8*(-32*exp(21) - 224*exp(15) - 224*exp(9) - 32*exp(3) + 4 + 112*exp(6) + 280*exp(12) + 112*exp(18) + 4*exp(
24))*exp(-24)/9 - x*exp(-x) + exp(3*x)

________________________________________________________________________________________