3.49.22 \(\int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+(3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5) \log (x)+(1250 x+3750 x^2+3750 x^3+1250 x^4) \log ^2(x)+(250 x+500 x^2+250 x^3) \log ^3(x)+(25 x+25 x^2) \log ^4(x)+x \log ^5(x)} \, dx\)

Optimal. Leaf size=17 \[ \frac {e^8 \log ^4(4)}{(5+5 x+\log (x))^4} \]

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Rubi [A]  time = 0.24, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 134, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {12, 6688, 6686} \begin {gather*} \frac {e^8 \log ^4(4)}{(5 x+\log (x)+5)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^8*(-4 - 20*x)*Log[4]^4)/(3125*x + 15625*x^2 + 31250*x^3 + 31250*x^4 + 15625*x^5 + 3125*x^6 + (3125*x +
12500*x^2 + 18750*x^3 + 12500*x^4 + 3125*x^5)*Log[x] + (1250*x + 3750*x^2 + 3750*x^3 + 1250*x^4)*Log[x]^2 + (2
50*x + 500*x^2 + 250*x^3)*Log[x]^3 + (25*x + 25*x^2)*Log[x]^4 + x*Log[x]^5),x]

[Out]

(E^8*Log[4]^4)/(5 + 5*x + Log[x])^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 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (e^8 \log ^4(4)\right ) \int \frac {-4-20 x}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx\\ &=\left (e^8 \log ^4(4)\right ) \int \frac {4 (-1-5 x)}{x (5+5 x+\log (x))^5} \, dx\\ &=\left (4 e^8 \log ^4(4)\right ) \int \frac {-1-5 x}{x (5+5 x+\log (x))^5} \, dx\\ &=\frac {e^8 \log ^4(4)}{(5+5 x+\log (x))^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {e^8 \log ^4(4)}{(5+5 x+\log (x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^8*(-4 - 20*x)*Log[4]^4)/(3125*x + 15625*x^2 + 31250*x^3 + 31250*x^4 + 15625*x^5 + 3125*x^6 + (312
5*x + 12500*x^2 + 18750*x^3 + 12500*x^4 + 3125*x^5)*Log[x] + (1250*x + 3750*x^2 + 3750*x^3 + 1250*x^4)*Log[x]^
2 + (250*x + 500*x^2 + 250*x^3)*Log[x]^3 + (25*x + 25*x^2)*Log[x]^4 + x*Log[x]^5),x]

[Out]

(E^8*Log[4]^4)/(5 + 5*x + Log[x])^4

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fricas [B]  time = 0.55, size = 74, normalized size = 4.35 \begin {gather*} \frac {16 \, e^{8} \log \relax (2)^{4}}{625 \, x^{4} + 20 \, {\left (x + 1\right )} \log \relax (x)^{3} + \log \relax (x)^{4} + 2500 \, x^{3} + 150 \, {\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} + 3750 \, x^{2} + 500 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \relax (x) + 2500 \, x + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*(-20*x-4)*exp(2)^4*log(2)^4/(x*log(x)^5+(25*x^2+25*x)*log(x)^4+(250*x^3+500*x^2+250*x)*log(x)^3+(
1250*x^4+3750*x^3+3750*x^2+1250*x)*log(x)^2+(3125*x^5+12500*x^4+18750*x^3+12500*x^2+3125*x)*log(x)+3125*x^6+15
625*x^5+31250*x^4+31250*x^3+15625*x^2+3125*x),x, algorithm="fricas")

[Out]

16*e^8*log(2)^4/(625*x^4 + 20*(x + 1)*log(x)^3 + log(x)^4 + 2500*x^3 + 150*(x^2 + 2*x + 1)*log(x)^2 + 3750*x^2
+ 500*(x^3 + 3*x^2 + 3*x + 1)*log(x) + 2500*x + 625)

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giac [B]  time = 0.16, size = 134, normalized size = 7.88 \begin {gather*} \frac {16 \, {\left (5 \, x + 1\right )} e^{8} \log \relax (2)^{4}}{3125 \, x^{5} + 2500 \, x^{4} \log \relax (x) + 750 \, x^{3} \log \relax (x)^{2} + 100 \, x^{2} \log \relax (x)^{3} + 5 \, x \log \relax (x)^{4} + 13125 \, x^{4} + 8000 \, x^{3} \log \relax (x) + 1650 \, x^{2} \log \relax (x)^{2} + 120 \, x \log \relax (x)^{3} + \log \relax (x)^{4} + 21250 \, x^{3} + 9000 \, x^{2} \log \relax (x) + 1050 \, x \log \relax (x)^{2} + 20 \, \log \relax (x)^{3} + 16250 \, x^{2} + 4000 \, x \log \relax (x) + 150 \, \log \relax (x)^{2} + 5625 \, x + 500 \, \log \relax (x) + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*(-20*x-4)*exp(2)^4*log(2)^4/(x*log(x)^5+(25*x^2+25*x)*log(x)^4+(250*x^3+500*x^2+250*x)*log(x)^3+(
1250*x^4+3750*x^3+3750*x^2+1250*x)*log(x)^2+(3125*x^5+12500*x^4+18750*x^3+12500*x^2+3125*x)*log(x)+3125*x^6+15
625*x^5+31250*x^4+31250*x^3+15625*x^2+3125*x),x, algorithm="giac")

[Out]

16*(5*x + 1)*e^8*log(2)^4/(3125*x^5 + 2500*x^4*log(x) + 750*x^3*log(x)^2 + 100*x^2*log(x)^3 + 5*x*log(x)^4 + 1
3125*x^4 + 8000*x^3*log(x) + 1650*x^2*log(x)^2 + 120*x*log(x)^3 + log(x)^4 + 21250*x^3 + 9000*x^2*log(x) + 105
0*x*log(x)^2 + 20*log(x)^3 + 16250*x^2 + 4000*x*log(x) + 150*log(x)^2 + 5625*x + 500*log(x) + 625)

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maple [A]  time = 0.03, size = 18, normalized size = 1.06

method	result	size
risch	\(\frac {16 \ln \relax (2)^{4} {\mathrm e}^{8}}{\left (5+\ln \relax (x )+5 x \right )^{4}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*(-20*x-4)*exp(2)^4*ln(2)^4/(x*ln(x)^5+(25*x^2+25*x)*ln(x)^4+(250*x^3+500*x^2+250*x)*ln(x)^3+(1250*x^4+3
750*x^3+3750*x^2+1250*x)*ln(x)^2+(3125*x^5+12500*x^4+18750*x^3+12500*x^2+3125*x)*ln(x)+3125*x^6+15625*x^5+3125
0*x^4+31250*x^3+15625*x^2+3125*x),x,method=_RETURNVERBOSE)

[Out]

16*ln(2)^4*exp(8)/(5+ln(x)+5*x)^4

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maxima [B]  time = 0.40, size = 74, normalized size = 4.35 \begin {gather*} \frac {16 \, e^{8} \log \relax (2)^{4}}{625 \, x^{4} + 20 \, {\left (x + 1\right )} \log \relax (x)^{3} + \log \relax (x)^{4} + 2500 \, x^{3} + 150 \, {\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} + 3750 \, x^{2} + 500 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \relax (x) + 2500 \, x + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*(-20*x-4)*exp(2)^4*log(2)^4/(x*log(x)^5+(25*x^2+25*x)*log(x)^4+(250*x^3+500*x^2+250*x)*log(x)^3+(
1250*x^4+3750*x^3+3750*x^2+1250*x)*log(x)^2+(3125*x^5+12500*x^4+18750*x^3+12500*x^2+3125*x)*log(x)+3125*x^6+15
625*x^5+31250*x^4+31250*x^3+15625*x^2+3125*x),x, algorithm="maxima")

[Out]

16*e^8*log(2)^4/(625*x^4 + 20*(x + 1)*log(x)^3 + log(x)^4 + 2500*x^3 + 150*(x^2 + 2*x + 1)*log(x)^2 + 3750*x^2
+ 500*(x^3 + 3*x^2 + 3*x + 1)*log(x) + 2500*x + 625)

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mupad [B]  time = 3.58, size = 17, normalized size = 1.00 \begin {gather*} \frac {16\,{\mathrm {e}}^8\,{\ln \relax (2)}^4}{{\left (5\,x+\ln \relax (x)+5\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*exp(8)*log(2)^4*(20*x + 4))/(3125*x + log(x)^4*(25*x + 25*x^2) + x*log(x)^5 + log(x)^3*(250*x + 500*x
^2 + 250*x^3) + log(x)*(3125*x + 12500*x^2 + 18750*x^3 + 12500*x^4 + 3125*x^5) + log(x)^2*(1250*x + 3750*x^2 +
3750*x^3 + 1250*x^4) + 15625*x^2 + 31250*x^3 + 31250*x^4 + 15625*x^5 + 3125*x^6),x)

[Out]

(16*exp(8)*log(2)^4)/(5*x + log(x) + 5)^4

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sympy [B]  time = 0.27, size = 78, normalized size = 4.59 \begin {gather*} \frac {16 e^{8} \log {\relax (2 )}^{4}}{625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + \left (20 x + 20\right ) \log {\relax (x )}^{3} + \left (150 x^{2} + 300 x + 150\right ) \log {\relax (x )}^{2} + \left (500 x^{3} + 1500 x^{2} + 1500 x + 500\right ) \log {\relax (x )} + \log {\relax (x )}^{4} + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*(-20*x-4)*exp(2)**4*ln(2)**4/(x*ln(x)**5+(25*x**2+25*x)*ln(x)**4+(250*x**3+500*x**2+250*x)*ln(x)*
*3+(1250*x**4+3750*x**3+3750*x**2+1250*x)*ln(x)**2+(3125*x**5+12500*x**4+18750*x**3+12500*x**2+3125*x)*ln(x)+3
125*x**6+15625*x**5+31250*x**4+31250*x**3+15625*x**2+3125*x),x)

[Out]

16*exp(8)*log(2)**4/(625*x**4 + 2500*x**3 + 3750*x**2 + 2500*x + (20*x + 20)*log(x)**3 + (150*x**2 + 300*x + 1
50)*log(x)**2 + (500*x**3 + 1500*x**2 + 1500*x + 500)*log(x) + log(x)**4 + 625)

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