3.6.25 \(\int \frac {e^{-4 x} (e^4 (-2430-1944 x)+e^{4+x} (1944 x^2+1944 x^3) \log (5)+e^{4+2 x} (-324 x^4-648 x^5) \log ^2(5)+e^{4+3 x} (-72 x^6+72 x^7) \log ^3(5)+18 e^{4+4 x} x^8 \log ^4(5))}{x^6} \, dx\)

Optimal. Leaf size=23 \[ 6 e^4 x^3 \left (-\frac {3 e^{-x}}{x^2}+\log (5)\right )^4 \]

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Rubi [B]  time = 1.78, antiderivative size = 86, normalized size of antiderivative = 3.74, number of steps used = 9, number of rules used = 6, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {6688, 12, 6742, 2197, 2176, 2194} \begin {gather*} \frac {486 e^{4-4 x}}{x^5}+6 e^4 x^3 \log ^4(5)-\frac {648 e^{4-3 x} \log (5)}{x^3}-72 e^{4-x} \log ^3(5)+72 e^{4-x} (1-x) \log ^3(5)+\frac {324 e^{4-2 x} \log ^2(5)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^4*(-2430 - 1944*x) + E^(4 + x)*(1944*x^2 + 1944*x^3)*Log[5] + E^(4 + 2*x)*(-324*x^4 - 648*x^5)*Log[5]^2
+ E^(4 + 3*x)*(-72*x^6 + 72*x^7)*Log[5]^3 + 18*E^(4 + 4*x)*x^8*Log[5]^4)/(E^(4*x)*x^6),x]

[Out]

(486*E^(4 - 4*x))/x^5 - (648*E^(4 - 3*x)*Log[5])/x^3 + (324*E^(4 - 2*x)*Log[5]^2)/x - 72*E^(4 - x)*Log[5]^3 +
72*E^(4 - x)*(1 - x)*Log[5]^3 + 6*E^4*x^3*Log[5]^4

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 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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} &=\int \frac {18 e^{4-4 x} \left (3-e^x x^2 \log (5)\right )^3 \left (-5-4 x-e^x x^2 \log (5)\right )}{x^6} \, dx\\ &=18 \int \frac {e^{4-4 x} \left (3-e^x x^2 \log (5)\right )^3 \left (-5-4 x-e^x x^2 \log (5)\right )}{x^6} \, dx\\ &=18 \int \left (-\frac {27 e^{4-4 x} (5+4 x)}{x^6}+\frac {108 e^{4-3 x} (1+x) \log (5)}{x^4}-\frac {18 e^{4-2 x} (1+2 x) \log ^2(5)}{x^2}+4 e^{4-x} (-1+x) \log ^3(5)+e^4 x^2 \log ^4(5)\right ) \, dx\\ &=6 e^4 x^3 \log ^4(5)-486 \int \frac {e^{4-4 x} (5+4 x)}{x^6} \, dx+(1944 \log (5)) \int \frac {e^{4-3 x} (1+x)}{x^4} \, dx-\left (324 \log ^2(5)\right ) \int \frac {e^{4-2 x} (1+2 x)}{x^2} \, dx+\left (72 \log ^3(5)\right ) \int e^{4-x} (-1+x) \, dx\\ &=\frac {486 e^{4-4 x}}{x^5}-\frac {648 e^{4-3 x} \log (5)}{x^3}+\frac {324 e^{4-2 x} \log ^2(5)}{x}+72 e^{4-x} (1-x) \log ^3(5)+6 e^4 x^3 \log ^4(5)+\left (72 \log ^3(5)\right ) \int e^{4-x} \, dx\\ &=\frac {486 e^{4-4 x}}{x^5}-\frac {648 e^{4-3 x} \log (5)}{x^3}+\frac {324 e^{4-2 x} \log ^2(5)}{x}-72 e^{4-x} \log ^3(5)+72 e^{4-x} (1-x) \log ^3(5)+6 e^4 x^3 \log ^4(5)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 25, normalized size = 1.09 \begin {gather*} \frac {6 e^{4-4 x} \left (-3+e^x x^2 \log (5)\right )^4}{x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(-2430 - 1944*x) + E^(4 + x)*(1944*x^2 + 1944*x^3)*Log[5] + E^(4 + 2*x)*(-324*x^4 - 648*x^5)*Lo
g[5]^2 + E^(4 + 3*x)*(-72*x^6 + 72*x^7)*Log[5]^3 + 18*E^(4 + 4*x)*x^8*Log[5]^4)/(E^(4*x)*x^6),x]

[Out]

(6*E^(4 - 4*x)*(-3 + E^x*x^2*Log[5])^4)/x^5

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fricas [B]  time = 0.56, size = 71, normalized size = 3.09 \begin {gather*} \frac {6 \, {\left (x^{8} e^{\left (4 \, x + 20\right )} \log \relax (5)^{4} - 12 \, x^{6} e^{\left (3 \, x + 20\right )} \log \relax (5)^{3} + 54 \, x^{4} e^{\left (2 \, x + 20\right )} \log \relax (5)^{2} - 108 \, x^{2} e^{\left (x + 20\right )} \log \relax (5) + 81 \, e^{20}\right )} e^{\left (-4 \, x - 16\right )}}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x^8*exp(4)*log(5)^4*exp(x)^4+(72*x^7-72*x^6)*exp(4)*log(5)^3*exp(x)^3+(-648*x^5-324*x^4)*exp(4)*
log(5)^2*exp(x)^2+(1944*x^3+1944*x^2)*exp(4)*log(5)*exp(x)+(-1944*x-2430)*exp(4))/x^6/exp(x)^4,x, algorithm="f
ricas")

[Out]

6*(x^8*e^(4*x + 20)*log(5)^4 - 12*x^6*e^(3*x + 20)*log(5)^3 + 54*x^4*e^(2*x + 20)*log(5)^2 - 108*x^2*e^(x + 20
)*log(5) + 81*e^20)*e^(-4*x - 16)/x^5

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giac [B]  time = 0.45, size = 67, normalized size = 2.91 \begin {gather*} \frac {6 \, {\left (x^{8} e^{4} \log \relax (5)^{4} - 12 \, x^{6} e^{\left (-x + 4\right )} \log \relax (5)^{3} + 54 \, x^{4} e^{\left (-2 \, x + 4\right )} \log \relax (5)^{2} - 108 \, x^{2} e^{\left (-3 \, x + 4\right )} \log \relax (5) + 81 \, e^{\left (-4 \, x + 4\right )}\right )}}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x^8*exp(4)*log(5)^4*exp(x)^4+(72*x^7-72*x^6)*exp(4)*log(5)^3*exp(x)^3+(-648*x^5-324*x^4)*exp(4)*
log(5)^2*exp(x)^2+(1944*x^3+1944*x^2)*exp(4)*log(5)*exp(x)+(-1944*x-2430)*exp(4))/x^6/exp(x)^4,x, algorithm="g
iac")

[Out]

6*(x^8*e^4*log(5)^4 - 12*x^6*e^(-x + 4)*log(5)^3 + 54*x^4*e^(-2*x + 4)*log(5)^2 - 108*x^2*e^(-3*x + 4)*log(5)
+ 81*e^(-4*x + 4))/x^5

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maple [B]  time = 0.05, size = 65, normalized size = 2.83

method	result	size
risch	\(6 x^{3} {\mathrm e}^{4} \ln \relax (5)^{4}-72 \ln \relax (5)^{3} x \,{\mathrm e}^{-x +4}+\frac {324 \ln \relax (5)^{2} {\mathrm e}^{4-2 x}}{x}-\frac {648 \ln \relax (5) {\mathrm e}^{4-3 x}}{x^{3}}+\frac {486 \,{\mathrm e}^{-4 x +4}}{x^{5}}\)	\(65\)
default	\(-2430 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x}}{5 x^{5}}+\frac {{\mathrm e}^{-4 x}}{5 x^{4}}-\frac {4 \,{\mathrm e}^{-4 x}}{15 x^{3}}+\frac {8 \,{\mathrm e}^{-4 x}}{15 x^{2}}-\frac {32 \,{\mathrm e}^{-4 x}}{15 x}+\frac {128 \expIntegralEi \left (1, 4 x \right )}{15}\right )-1944 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x}}{4 x^{4}}+\frac {{\mathrm e}^{-4 x}}{3 x^{3}}-\frac {2 \,{\mathrm e}^{-4 x}}{3 x^{2}}+\frac {8 \,{\mathrm e}^{-4 x}}{3 x}-\frac {32 \expIntegralEi \left (1, 4 x \right )}{3}\right )+6 x^{3} {\mathrm e}^{4} \ln \relax (5)^{4}-\frac {648 \,{\mathrm e}^{-3 x} {\mathrm e}^{4} \ln \relax (5)}{x^{3}}+\frac {324 \,{\mathrm e}^{-2 x} {\mathrm e}^{4} \ln \relax (5)^{2}}{x}-72 x \,{\mathrm e}^{-x} {\mathrm e}^{4} \ln \relax (5)^{3}\)	\(159\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x^8*exp(4)*ln(5)^4*exp(x)^4+(72*x^7-72*x^6)*exp(4)*ln(5)^3*exp(x)^3+(-648*x^5-324*x^4)*exp(4)*ln(5)^2*
exp(x)^2+(1944*x^3+1944*x^2)*exp(4)*ln(5)*exp(x)+(-1944*x-2430)*exp(4))/x^6/exp(x)^4,x,method=_RETURNVERBOSE)

[Out]

6*x^3*exp(4)*ln(5)^4-72*ln(5)^3*x*exp(-x+4)+324/x*ln(5)^2*exp(4-2*x)-648*ln(5)/x^3*exp(4-3*x)+486/x^5*exp(-4*x
+4)

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maxima [C]  time = 0.54, size = 106, normalized size = 4.61 \begin {gather*} 6 \, x^{3} e^{4} \log \relax (5)^{4} - 72 \, {\left (x e^{4} + e^{4}\right )} e^{\left (-x\right )} \log \relax (5)^{3} - 648 \, {\rm Ei}\left (-2 \, x\right ) e^{4} \log \relax (5)^{2} + 648 \, e^{4} \Gamma \left (-1, 2 \, x\right ) \log \relax (5)^{2} + 72 \, e^{\left (-x + 4\right )} \log \relax (5)^{3} - 17496 \, e^{4} \Gamma \left (-2, 3 \, x\right ) \log \relax (5) - 52488 \, e^{4} \Gamma \left (-3, 3 \, x\right ) \log \relax (5) + 497664 \, e^{4} \Gamma \left (-4, 4 \, x\right ) + 2488320 \, e^{4} \Gamma \left (-5, 4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x^8*exp(4)*log(5)^4*exp(x)^4+(72*x^7-72*x^6)*exp(4)*log(5)^3*exp(x)^3+(-648*x^5-324*x^4)*exp(4)*
log(5)^2*exp(x)^2+(1944*x^3+1944*x^2)*exp(4)*log(5)*exp(x)+(-1944*x-2430)*exp(4))/x^6/exp(x)^4,x, algorithm="m
axima")

[Out]

6*x^3*e^4*log(5)^4 - 72*(x*e^4 + e^4)*e^(-x)*log(5)^3 - 648*Ei(-2*x)*e^4*log(5)^2 + 648*e^4*gamma(-1, 2*x)*log
(5)^2 + 72*e^(-x + 4)*log(5)^3 - 17496*e^4*gamma(-2, 3*x)*log(5) - 52488*e^4*gamma(-3, 3*x)*log(5) + 497664*e^
4*gamma(-4, 4*x) + 2488320*e^4*gamma(-5, 4*x)

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mupad [B]  time = 0.36, size = 64, normalized size = 2.78 \begin {gather*} \frac {486\,{\mathrm {e}}^{4-4\,x}}{x^5}-72\,x\,{\mathrm {e}}^{4-x}\,{\ln \relax (5)}^3-\frac {648\,{\mathrm {e}}^{4-3\,x}\,\ln \relax (5)}{x^3}+6\,x^3\,{\mathrm {e}}^4\,{\ln \relax (5)}^4+\frac {324\,{\mathrm {e}}^{4-2\,x}\,{\ln \relax (5)}^2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4*x)*(exp(4)*(1944*x + 2430) + exp(3*x)*exp(4)*log(5)^3*(72*x^6 - 72*x^7) + exp(2*x)*exp(4)*log(5)^
2*(324*x^4 + 648*x^5) - 18*x^8*exp(4*x)*exp(4)*log(5)^4 - exp(4)*exp(x)*log(5)*(1944*x^2 + 1944*x^3)))/x^6,x)

[Out]

(486*exp(4 - 4*x))/x^5 - 72*x*exp(4 - x)*log(5)^3 - (648*exp(4 - 3*x)*log(5))/x^3 + 6*x^3*exp(4)*log(5)^4 + (3
24*exp(4 - 2*x)*log(5)^2)/x

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sympy [B]  time = 0.33, size = 82, normalized size = 3.57 \begin {gather*} 6 x^{3} e^{4} \log {\relax (5 )}^{4} + \frac {- 72 x^{10} e^{4} e^{- x} \log {\relax (5 )}^{3} + 324 x^{8} e^{4} e^{- 2 x} \log {\relax (5 )}^{2} - 648 x^{6} e^{4} e^{- 3 x} \log {\relax (5 )} + 486 x^{4} e^{4} e^{- 4 x}}{x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x**8*exp(4)*ln(5)**4*exp(x)**4+(72*x**7-72*x**6)*exp(4)*ln(5)**3*exp(x)**3+(-648*x**5-324*x**4)*
exp(4)*ln(5)**2*exp(x)**2+(1944*x**3+1944*x**2)*exp(4)*ln(5)*exp(x)+(-1944*x-2430)*exp(4))/x**6/exp(x)**4,x)

[Out]

6*x**3*exp(4)*log(5)**4 + (-72*x**10*exp(4)*exp(-x)*log(5)**3 + 324*x**8*exp(4)*exp(-2*x)*log(5)**2 - 648*x**6
*exp(4)*exp(-3*x)*log(5) + 486*x**4*exp(4)*exp(-4*x))/x**9

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