3.25.4 \(\int \frac {(6-2 e^x x)^5 (-6 e^x x+e^{2 x} (7 x^2+5 x^3))+(-6 x+3 x^2) \log (4)+e^x (2 x^2-x^3) \log (4)}{-27 e^x+9 e^{2 x} x} \, dx\) [2404]

Optimal. Leaf size=28 \[ \frac {1}{9} x^2 \left (32 \left (3-e^x x\right )^5+e^{-x} \log (4)\right ) \]

[Out]

1/9*((-2*exp(x)*x+6)^5+2*ln(2)/exp(x))*x^2

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(28)=56\).
time = 1.61, antiderivative size = 72, normalized size of antiderivative = 2.57, number of steps used = 77, number of rules used = 7, integrand size = 82, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {6873, 12, 6820, 6874, 2227, 2207, 2225} \begin {gather*} -\frac {32}{9} e^{5 x} x^7+\frac {160}{3} e^{4 x} x^6-320 e^{3 x} x^5+960 e^{2 x} x^4-1440 e^x x^3+864 x^2+\frac {1}{9} e^{-x} x^2 \log (4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((6 - 2*E^x*x)^5*(-6*E^x*x + E^(2*x)*(7*x^2 + 5*x^3)) + (-6*x + 3*x^2)*Log[4] + E^x*(2*x^2 - x^3)*Log[4])/
(-27*E^x + 9*E^(2*x)*x),x]

[Out]

864*x^2 - 1440*E^x*x^3 + 960*E^(2*x)*x^4 - 320*E^(3*x)*x^5 + (160*E^(4*x)*x^6)/3 - (32*E^(5*x)*x^7)/9 + (x^2*L
og[4])/(9*E^x)

Rule 12

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

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 2227

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] &&  !TrueQ[$UseGamma]

Rule 6820

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

Rule 6873

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

Rule 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (-\left (6-2 e^x x\right )^5 \left (-6 e^x x+e^{2 x} \left (7 x^2+5 x^3\right )\right )-\left (-6 x+3 x^2\right ) \log (4)-e^x \left (2 x^2-x^3\right ) \log (4)\right )}{9 \left (3-e^x x\right )} \, dx\\ &=\frac {1}{9} \int \frac {e^{-x} \left (-\left (6-2 e^x x\right )^5 \left (-6 e^x x+e^{2 x} \left (7 x^2+5 x^3\right )\right )-\left (-6 x+3 x^2\right ) \log (4)-e^x \left (2 x^2-x^3\right ) \log (4)\right )}{3-e^x x} \, dx\\ &=\frac {1}{9} \int \frac {e^{-x} x \left (32 e^x \left (-3+e^x x\right )^5 \left (-6+e^x x (7+5 x)\right )-3 (-2+x) \log (4)+e^x (-2+x) x \log (4)\right )}{3-e^x x} \, dx\\ &=\frac {1}{9} \int \left (15552 x+17280 e^{2 x} x^3 (2+x)-12960 e^x x^2 (3+x)+960 e^{4 x} x^5 (3+2 x)-2880 e^{3 x} x^4 (5+3 x)-32 e^{5 x} x^6 (7+5 x)-e^{-x} (-2+x) x \log (4)\right ) \, dx\\ &=864 x^2-\frac {32}{9} \int e^{5 x} x^6 (7+5 x) \, dx+\frac {320}{3} \int e^{4 x} x^5 (3+2 x) \, dx-320 \int e^{3 x} x^4 (5+3 x) \, dx-1440 \int e^x x^2 (3+x) \, dx+1920 \int e^{2 x} x^3 (2+x) \, dx-\frac {1}{9} \log (4) \int e^{-x} (-2+x) x \, dx\\ &=864 x^2-\frac {32}{9} \int \left (7 e^{5 x} x^6+5 e^{5 x} x^7\right ) \, dx+\frac {320}{3} \int \left (3 e^{4 x} x^5+2 e^{4 x} x^6\right ) \, dx-320 \int \left (5 e^{3 x} x^4+3 e^{3 x} x^5\right ) \, dx-1440 \int \left (3 e^x x^2+e^x x^3\right ) \, dx+1920 \int \left (2 e^{2 x} x^3+e^{2 x} x^4\right ) \, dx-\frac {1}{9} \log (4) \int \left (-2 e^{-x} x+e^{-x} x^2\right ) \, dx\\ &=864 x^2-\frac {160}{9} \int e^{5 x} x^7 \, dx-\frac {224}{9} \int e^{5 x} x^6 \, dx+\frac {640}{3} \int e^{4 x} x^6 \, dx+320 \int e^{4 x} x^5 \, dx-960 \int e^{3 x} x^5 \, dx-1440 \int e^x x^3 \, dx-1600 \int e^{3 x} x^4 \, dx+1920 \int e^{2 x} x^4 \, dx+3840 \int e^{2 x} x^3 \, dx-4320 \int e^x x^2 \, dx-\frac {1}{9} \log (4) \int e^{-x} x^2 \, dx+\frac {1}{9} (2 \log (4)) \int e^{-x} x \, dx\\ &=864 x^2-4320 e^x x^2-1440 e^x x^3+1920 e^{2 x} x^3+960 e^{2 x} x^4-\frac {1600}{3} e^{3 x} x^4-320 e^{3 x} x^5+80 e^{4 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {224}{45} e^{5 x} x^6-\frac {32}{9} e^{5 x} x^7-\frac {2}{9} e^{-x} x \log (4)+\frac {1}{9} e^{-x} x^2 \log (4)+\frac {224}{9} \int e^{5 x} x^6 \, dx+\frac {448}{15} \int e^{5 x} x^5 \, dx-320 \int e^{4 x} x^5 \, dx-400 \int e^{4 x} x^4 \, dx+1600 \int e^{3 x} x^4 \, dx+\frac {6400}{3} \int e^{3 x} x^3 \, dx-3840 \int e^{2 x} x^3 \, dx+4320 \int e^x x^2 \, dx-5760 \int e^{2 x} x^2 \, dx+8640 \int e^x x \, dx+\frac {1}{9} (2 \log (4)) \int e^{-x} \, dx-\frac {1}{9} (2 \log (4)) \int e^{-x} x \, dx\\ &=8640 e^x x+864 x^2-2880 e^{2 x} x^2-1440 e^x x^3+\frac {6400}{9} e^{3 x} x^3+960 e^{2 x} x^4-100 e^{4 x} x^4-320 e^{3 x} x^5+\frac {448}{75} e^{5 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7-\frac {2}{9} e^{-x} \log (4)+\frac {1}{9} e^{-x} x^2 \log (4)-\frac {448}{15} \int e^{5 x} x^4 \, dx-\frac {448}{15} \int e^{5 x} x^5 \, dx+400 \int e^{4 x} x^3 \, dx+400 \int e^{4 x} x^4 \, dx-\frac {6400}{3} \int e^{3 x} x^2 \, dx-\frac {6400}{3} \int e^{3 x} x^3 \, dx+5760 \int e^{2 x} x \, dx+5760 \int e^{2 x} x^2 \, dx-8640 \int e^x \, dx-8640 \int e^x x \, dx-\frac {1}{9} (2 \log (4)) \int e^{-x} \, dx\\ &=-8640 e^x+2880 e^{2 x} x+864 x^2-\frac {6400}{9} e^{3 x} x^2-1440 e^x x^3+100 e^{4 x} x^3+960 e^{2 x} x^4-\frac {448}{75} e^{5 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)+\frac {1792}{75} \int e^{5 x} x^3 \, dx+\frac {448}{15} \int e^{5 x} x^4 \, dx-300 \int e^{4 x} x^2 \, dx-400 \int e^{4 x} x^3 \, dx+\frac {12800}{9} \int e^{3 x} x \, dx+\frac {6400}{3} \int e^{3 x} x^2 \, dx-2880 \int e^{2 x} \, dx-5760 \int e^{2 x} x \, dx+8640 \int e^x \, dx\\ &=-1440 e^{2 x}+\frac {12800}{27} e^{3 x} x+864 x^2-75 e^{4 x} x^2-1440 e^x x^3+\frac {1792}{375} e^{5 x} x^3+960 e^{2 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)-\frac {1792}{125} \int e^{5 x} x^2 \, dx-\frac {1792}{75} \int e^{5 x} x^3 \, dx+150 \int e^{4 x} x \, dx+300 \int e^{4 x} x^2 \, dx-\frac {12800}{27} \int e^{3 x} \, dx-\frac {12800}{9} \int e^{3 x} x \, dx+2880 \int e^{2 x} \, dx\\ &=-\frac {12800 e^{3 x}}{81}+\frac {75}{2} e^{4 x} x+864 x^2-\frac {1792}{625} e^{5 x} x^2-1440 e^x x^3+960 e^{2 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)+\frac {3584}{625} \int e^{5 x} x \, dx+\frac {1792}{125} \int e^{5 x} x^2 \, dx-\frac {75}{2} \int e^{4 x} \, dx-150 \int e^{4 x} x \, dx+\frac {12800}{27} \int e^{3 x} \, dx\\ &=-\frac {75 e^{4 x}}{8}+\frac {3584 e^{5 x} x}{3125}+864 x^2-1440 e^x x^3+960 e^{2 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)-\frac {3584 \int e^{5 x} \, dx}{3125}-\frac {3584}{625} \int e^{5 x} x \, dx+\frac {75}{2} \int e^{4 x} \, dx\\ &=-\frac {3584 e^{5 x}}{15625}+864 x^2-1440 e^x x^3+960 e^{2 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)+\frac {3584 \int e^{5 x} \, dx}{3125}\\ &=864 x^2-1440 e^x x^3+960 e^{2 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(28)=56\).
time = 0.29, size = 63, normalized size = 2.25 \begin {gather*} \frac {1}{9} x^2 \left (7776-12960 e^x x+8640 e^{2 x} x^2-2880 e^{3 x} x^3+480 e^{4 x} x^4-32 e^{5 x} x^5+e^{-x} \log (4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((6 - 2*E^x*x)^5*(-6*E^x*x + E^(2*x)*(7*x^2 + 5*x^3)) + (-6*x + 3*x^2)*Log[4] + E^x*(2*x^2 - x^3)*Lo
g[4])/(-27*E^x + 9*E^(2*x)*x),x]

[Out]

(x^2*(7776 - 12960*E^x*x + 8640*E^(2*x)*x^2 - 2880*E^(3*x)*x^3 + 480*E^(4*x)*x^4 - 32*E^(5*x)*x^5 + Log[4]/E^x
))/9

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs. \(2(23)=46\).
time = 0.10, size = 61, normalized size = 2.18

method	result	size
risch	\(864 x^{2}-\frac {32 x^{7} {\mathrm e}^{5 x}}{9}+\frac {160 x^{6} {\mathrm e}^{4 x}}{3}-320 x^{5} {\mathrm e}^{3 x}+960 \,{\mathrm e}^{2 x} x^{4}-1440 \,{\mathrm e}^{x} x^{3}+\frac {2 x^{2} \ln \left (2\right ) {\mathrm e}^{-x}}{9}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((5*x^3+7*x^2)*exp(x)^2-6*exp(x)*x)*(-2*exp(x)*x+6)^5+2*(-x^3+2*x^2)*ln(2)*exp(x)+2*(3*x^2-6*x)*ln(2))/(9
*x*exp(x)^2-27*exp(x)),x,method=_RETURNVERBOSE)

[Out]

864*x^2-32/9*x^7*exp(5*x)+160/3*x^6*exp(4*x)-320*x^5*exp(3*x)+960*exp(2*x)*x^4-1440*exp(x)*x^3+2/9*x^2*ln(2)*e
xp(-x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
time = 0.50, size = 60, normalized size = 2.14 \begin {gather*} -\frac {32}{9} \, x^{7} e^{\left (5 \, x\right )} + \frac {160}{3} \, x^{6} e^{\left (4 \, x\right )} - 320 \, x^{5} e^{\left (3 \, x\right )} + 960 \, x^{4} e^{\left (2 \, x\right )} - 1440 \, x^{3} e^{x} + \frac {2}{9} \, x^{2} e^{\left (-x\right )} \log \left (2\right ) + 864 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3+7*x^2)*exp(x)^2-6*exp(x)*x)*(-2*exp(x)*x+6)^5+2*(-x^3+2*x^2)*log(2)*exp(x)+2*(3*x^2-6*x)*lo
g(2))/(9*x*exp(x)^2-27*exp(x)),x, algorithm="maxima")

[Out]

-32/9*x^7*e^(5*x) + 160/3*x^6*e^(4*x) - 320*x^5*e^(3*x) + 960*x^4*e^(2*x) - 1440*x^3*e^x + 2/9*x^2*e^(-x)*log(
2) + 864*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
time = 0.36, size = 66, normalized size = 2.36 \begin {gather*} -\frac {2}{9} \, {\left (16 \, x^{7} e^{\left (6 \, x\right )} - 240 \, x^{6} e^{\left (5 \, x\right )} + 1440 \, x^{5} e^{\left (4 \, x\right )} - 4320 \, x^{4} e^{\left (3 \, x\right )} + 6480 \, x^{3} e^{\left (2 \, x\right )} - 3888 \, x^{2} e^{x} - x^{2} \log \left (2\right )\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3+7*x^2)*exp(x)^2-6*exp(x)*x)*(-2*exp(x)*x+6)^5+2*(-x^3+2*x^2)*log(2)*exp(x)+2*(3*x^2-6*x)*lo
g(2))/(9*x*exp(x)^2-27*exp(x)),x, algorithm="fricas")

[Out]

-2/9*(16*x^7*e^(6*x) - 240*x^6*e^(5*x) + 1440*x^5*e^(4*x) - 4320*x^4*e^(3*x) + 6480*x^3*e^(2*x) - 3888*x^2*e^x
- x^2*log(2))*e^(-x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).
time = 0.16, size = 70, normalized size = 2.50 \begin {gather*} - \frac {32 x^{7} e^{5 x}}{9} + \frac {160 x^{6} e^{4 x}}{3} - 320 x^{5} e^{3 x} + 960 x^{4} e^{2 x} - 1440 x^{3} e^{x} + 864 x^{2} + \frac {2 x^{2} e^{- x} \log {\left (2 \right )}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x**3+7*x**2)*exp(x)**2-6*exp(x)*x)*(-2*exp(x)*x+6)**5+2*(-x**3+2*x**2)*ln(2)*exp(x)+2*(3*x**2-6
*x)*ln(2))/(9*x*exp(x)**2-27*exp(x)),x)

[Out]

-32*x**7*exp(5*x)/9 + 160*x**6*exp(4*x)/3 - 320*x**5*exp(3*x) + 960*x**4*exp(2*x) - 1440*x**3*exp(x) + 864*x**
2 + 2*x**2*exp(-x)*log(2)/9

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
time = 0.41, size = 66, normalized size = 2.36 \begin {gather*} -\frac {2}{9} \, {\left (16 \, x^{7} e^{\left (6 \, x\right )} - 240 \, x^{6} e^{\left (5 \, x\right )} + 1440 \, x^{5} e^{\left (4 \, x\right )} - 4320 \, x^{4} e^{\left (3 \, x\right )} + 6480 \, x^{3} e^{\left (2 \, x\right )} - 3888 \, x^{2} e^{x} - x^{2} \log \left (2\right )\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3+7*x^2)*exp(x)^2-6*exp(x)*x)*(-2*exp(x)*x+6)^5+2*(-x^3+2*x^2)*log(2)*exp(x)+2*(3*x^2-6*x)*lo
g(2))/(9*x*exp(x)^2-27*exp(x)),x, algorithm="giac")

[Out]

-2/9*(16*x^7*e^(6*x) - 240*x^6*e^(5*x) + 1440*x^5*e^(4*x) - 4320*x^4*e^(3*x) + 6480*x^3*e^(2*x) - 3888*x^2*e^x
- x^2*log(2))*e^(-x)

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Mupad [B]
time = 1.43, size = 55, normalized size = 1.96 \begin {gather*} \frac {2\,x^2\,\left ({\mathrm {e}}^{-x}\,\ln \left (2\right )+4320\,x^2\,{\mathrm {e}}^{2\,x}-1440\,x^3\,{\mathrm {e}}^{3\,x}+240\,x^4\,{\mathrm {e}}^{4\,x}-16\,x^5\,{\mathrm {e}}^{5\,x}-6480\,x\,{\mathrm {e}}^x+3888\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2)*(6*x - 3*x^2) + (2*x*exp(x) - 6)^5*(exp(2*x)*(7*x^2 + 5*x^3) - 6*x*exp(x)) - 2*exp(x)*log(2)*(2*
x^2 - x^3))/(27*exp(x) - 9*x*exp(2*x)),x)

[Out]

(2*x^2*(exp(-x)*log(2) + 4320*x^2*exp(2*x) - 1440*x^3*exp(3*x) + 240*x^4*exp(4*x) - 16*x^5*exp(5*x) - 6480*x*e
xp(x) + 3888))/9

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