\(\int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+(3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5) \log (x)+(1250 x+3750 x^2+3750 x^3+1250 x^4) \log ^2(x)+(250 x+500 x^2+250 x^3) \log ^3(x)+(25 x+25 x^2) \log ^4(x)+x \log ^5(x)} \, dx\) [4822]

Optimal result

Integrand size = 134, antiderivative size = 17 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {e^8 \log ^4(4)}{(5+5 x+\log (x))^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {12, 6820, 6818} \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {e^8 \log ^4(4)}{(5 x+\log (x)+5)^4} \]

[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 6818

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 6820

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

Rubi steps \begin{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {e^8 \log ^4(4)}{(5+5 x+\log (x))^4} \]

[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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06

method	result	size
risch	\(\frac {16 \ln \left (2\right )^{4} {\mathrm e}^{8}}{\left (5+\ln \left (x \right )+5 x \right )^{4}}\)	\(18\)
default	\(\frac {16 \ln \left (2\right )^{4} {\mathrm e}^{8}}{\left (5+\ln \left (x \right )+5 x \right )^{4}}\)	\(20\)
parallelrisch	\(\frac {16 \,{\mathrm e}^{8} \ln \left (2\right )^{4}}{625 x^{4}+500 x^{3} \ln \left (x \right )+150 x^{2} \ln \left (x \right )^{2}+20 x \ln \left (x \right )^{3}+\ln \left (x \right )^{4}+2500 x^{3}+1500 x^{2} \ln \left (x \right )+300 x \ln \left (x \right )^{2}+20 \ln \left (x \right )^{3}+3750 x^{2}+1500 x \ln \left (x \right )+150 \ln \left (x \right )^{2}+2500 x +500 \ln \left (x \right )+625}\)	\(95\)

[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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (17) = 34\).

Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 4.35 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16 \, e^{8} \log \left (2\right )^{4}}{625 \, x^{4} + 20 \, {\left (x + 1\right )} \log \left (x\right )^{3} + \log \left (x\right )^{4} + 2500 \, x^{3} + 150 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 3750 \, x^{2} + 500 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x\right ) + 2500 \, x + 625} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (19) = 38\).

Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 4.59 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16 e^{8} \log {\left (2 \right )}^{4}}{625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + \left (20 x + 20\right ) \log {\left (x \right )}^{3} + \left (150 x^{2} + 300 x + 150\right ) \log {\left (x \right )}^{2} + \left (500 x^{3} + 1500 x^{2} + 1500 x + 500\right ) \log {\left (x \right )} + \log {\left (x \right )}^{4} + 625} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (17) = 34\).

Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 4.35 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16 \, e^{8} \log \left (2\right )^{4}}{625 \, x^{4} + 20 \, {\left (x + 1\right )} \log \left (x\right )^{3} + \log \left (x\right )^{4} + 2500 \, x^{3} + 150 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 3750 \, x^{2} + 500 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x\right ) + 2500 \, x + 625} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (17) = 34\).

Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 7.88 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16 \, {\left (5 \, x + 1\right )} e^{8} \log \left (2\right )^{4}}{3125 \, x^{5} + 2500 \, x^{4} \log \left (x\right ) + 750 \, x^{3} \log \left (x\right )^{2} + 100 \, x^{2} \log \left (x\right )^{3} + 5 \, x \log \left (x\right )^{4} + 13125 \, x^{4} + 8000 \, x^{3} \log \left (x\right ) + 1650 \, x^{2} \log \left (x\right )^{2} + 120 \, x \log \left (x\right )^{3} + \log \left (x\right )^{4} + 21250 \, x^{3} + 9000 \, x^{2} \log \left (x\right ) + 1050 \, x \log \left (x\right )^{2} + 20 \, \log \left (x\right )^{3} + 16250 \, x^{2} + 4000 \, x \log \left (x\right ) + 150 \, \log \left (x\right )^{2} + 5625 \, x + 500 \, \log \left (x\right ) + 625} \]

[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)

Mupad [B] (verification not implemented)

Time = 11.47 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16\,{\mathrm {e}}^8\,{\ln \left (2\right )}^4}{{\left (5\,x+\ln \left (x\right )+5\right )}^4} \]

[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