Integrand size = 112, antiderivative size = 17 \[ \int \frac {\left (-157464-96228 e-9396 e^2-332 e^3-4 e^4\right ) \log (2)+\left (-87480-50220 e-3360 e^2-60 e^3\right ) \log (2) \log (x)+\left (-16200-8700 e-300 e^2\right ) \log (2) \log ^2(x)+(-1000-500 e) \log (2) \log ^3(x)}{3125 x+3125 x \log (x)+1250 x \log ^2(x)+250 x \log ^3(x)+25 x \log ^4(x)+x \log ^5(x)} \, dx=\log (2) \left (5+\frac {2+e}{5+\log (x)}\right )^4 \] Output:
(5+(exp(1)+2)/(5+ln(x)))^4*ln(2)
Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(17)=34\).
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.82 \[ \int \frac {\left (-157464-96228 e-9396 e^2-332 e^3-4 e^4\right ) \log (2)+\left (-87480-50220 e-3360 e^2-60 e^3\right ) \log (2) \log (x)+\left (-16200-8700 e-300 e^2\right ) \log (2) \log ^2(x)+(-1000-500 e) \log (2) \log ^3(x)}{3125 x+3125 x \log (x)+1250 x \log ^2(x)+250 x \log ^3(x)+25 x \log ^4(x)+x \log ^5(x)} \, dx=\frac {(2+e) \log (2) \left ((2+e)^3+20 (2+e)^2 (5+\log (x))+150 (2+e) (5+\log (x))^2+500 (5+\log (x))^3\right )}{(5+\log (x))^4} \] Input:
Integrate[((-157464 - 96228*E - 9396*E^2 - 332*E^3 - 4*E^4)*Log[2] + (-874 80 - 50220*E - 3360*E^2 - 60*E^3)*Log[2]*Log[x] + (-16200 - 8700*E - 300*E ^2)*Log[2]*Log[x]^2 + (-1000 - 500*E)*Log[2]*Log[x]^3)/(3125*x + 3125*x*Lo g[x] + 1250*x*Log[x]^2 + 250*x*Log[x]^3 + 25*x*Log[x]^4 + x*Log[x]^5),x]
Output:
((2 + E)*Log[2]*((2 + E)^3 + 20*(2 + E)^2*(5 + Log[x]) + 150*(2 + E)*(5 + Log[x])^2 + 500*(5 + Log[x])^3))/(5 + Log[x])^4
Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3039, 6, 6, 6, 6, 6, 6, 27, 2006, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(-1000-500 e) \log (2) \log ^3(x)+\left (-16200-8700 e-300 e^2\right ) \log (2) \log ^2(x)+\left (-87480-50220 e-3360 e^2-60 e^3\right ) \log (2) \log (x)+\left (-157464-96228 e-9396 e^2-332 e^3-4 e^4\right ) \log (2)}{3125 x+x \log ^5(x)+25 x \log ^4(x)+250 x \log ^3(x)+1250 x \log ^2(x)+3125 x \log (x)} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \int -\frac {4 \left (125 e \log (2) \log ^3(x)+250 \log (2) \log ^3(x)+75 e^2 \log (2) \log ^2(x)+2175 e \log (2) \log ^2(x)+4050 \log (2) \log ^2(x)+15 e^3 \log (2) \log (x)+840 e^2 \log (2) \log (x)+12555 e \log (2) \log (x)+21870 \log (2) \log (x)+(2+e) (27+e)^3 \log (2)\right )}{(\log (x)+5)^5}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {4 \left (125 e \log (2) \log ^3(x)+250 \log (2) \log ^3(x)+75 e^2 \log (2) \log ^2(x)+2175 e \log (2) \log ^2(x)+4050 \log (2) \log ^2(x)+(21870+12555 e) \log (2) \log (x)+15 e^3 \log (2) \log (x)+840 e^2 \log (2) \log (x)+(2+e) (27+e)^3 \log (2)\right )}{(\log (x)+5)^5}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {4 \left (125 e \log (2) \log ^3(x)+250 \log (2) \log ^3(x)+75 e^2 \log (2) \log ^2(x)+2175 e \log (2) \log ^2(x)+4050 \log (2) \log ^2(x)+\left (840 e^2+15 e^3\right ) \log (2) \log (x)+(21870+12555 e) \log (2) \log (x)+(2+e) (27+e)^3 \log (2)\right )}{(\log (x)+5)^5}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {4 \left (125 e \log (2) \log ^3(x)+250 \log (2) \log ^3(x)+75 e^2 \log (2) \log ^2(x)+2175 e \log (2) \log ^2(x)+4050 \log (2) \log ^2(x)+\left (21870+12555 e+840 e^2+15 e^3\right ) \log (2) \log (x)+(2+e) (27+e)^3 \log (2)\right )}{(\log (x)+5)^5}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {4 \left (125 e \log (2) \log ^3(x)+250 \log (2) \log ^3(x)+(4050+2175 e) \log (2) \log ^2(x)+75 e^2 \log (2) \log ^2(x)+\left (21870+12555 e+840 e^2+15 e^3\right ) \log (2) \log (x)+(2+e) (27+e)^3 \log (2)\right )}{(\log (x)+5)^5}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {4 \left (125 e \log (2) \log ^3(x)+250 \log (2) \log ^3(x)+\left (4050+2175 e+75 e^2\right ) \log (2) \log ^2(x)+\left (21870+12555 e+840 e^2+15 e^3\right ) \log (2) \log (x)+(2+e) (27+e)^3 \log (2)\right )}{(\log (x)+5)^5}d\log (x)\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int -\frac {4 \left ((250+125 e) \log (2) \log ^3(x)+\left (4050+2175 e+75 e^2\right ) \log (2) \log ^2(x)+\left (21870+12555 e+840 e^2+15 e^3\right ) \log (2) \log (x)+(2+e) (27+e)^3 \log (2)\right )}{(\log (x)+5)^5}d\log (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 \int \frac {125 (2+e) \log (2) \log ^3(x)+75 (2+e) (27+e) \log (2) \log ^2(x)+15 (2+e) (27+e)^2 \log (2) \log (x)+(2+e) (27+e)^3 \log (2)}{(\log (x)+5)^5}d\log (x)\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle -4 \int \frac {\left (5 \sqrt [3]{(2+e) \log (2)} \log (x)+(27+e) \sqrt [3]{(2+e) \log (2)}\right )^3}{(\log (x)+5)^5}d\log (x)\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\log (2) (5 \log (x)+e+27)^4}{(\log (x)+5)^4}\) |
Input:
Int[((-157464 - 96228*E - 9396*E^2 - 332*E^3 - 4*E^4)*Log[2] + (-87480 - 5 0220*E - 3360*E^2 - 60*E^3)*Log[2]*Log[x] + (-16200 - 8700*E - 300*E^2)*Lo g[2]*Log[x]^2 + (-1000 - 500*E)*Log[2]*Log[x]^3)/(3125*x + 3125*x*Log[x] + 1250*x*Log[x]^2 + 250*x*Log[x]^3 + 25*x*Log[x]^4 + x*Log[x]^5),x]
Output:
(Log[2]*(27 + E + 5*Log[x])^4)/(5 + Log[x])^4
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(18)=36\).
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 4.12
| method | result | size |
| default | \(-4 \ln \left (2\right ) \left ({\mathrm e}+2\right ) \left (-\frac {15 \,{\mathrm e}^{2}+60 \,{\mathrm e}+60}{3 \left (5+\ln \left (x \right )\right )^{3}}-\frac {{\mathrm e}^{3}+6 \,{\mathrm e}^{2}+12 \,{\mathrm e}+8}{4 \left (5+\ln \left (x \right )\right )^{4}}-\frac {125}{5+\ln \left (x \right )}-\frac {75 \,{\mathrm e}+150}{2 \left (5+\ln \left (x \right )\right )^{2}}\right )\) | \(70\) |
| risch | \(\frac {\ln \left (2\right ) \left ({\mathrm e}^{4}+20 \ln \left (x \right ) {\mathrm e}^{3}+150 \,{\mathrm e}^{2} \ln \left (x \right )^{2}+500 \ln \left (x \right )^{3} {\mathrm e}+108 \,{\mathrm e}^{3}+1620 \,{\mathrm e}^{2} \ln \left (x \right )+8100 \,{\mathrm e} \ln \left (x \right )^{2}+1000 \ln \left (x \right )^{3}+4374 \,{\mathrm e}^{2}+43740 \,{\mathrm e} \ln \left (x \right )+15600 \ln \left (x \right )^{2}+78732 \,{\mathrm e}+81160 \ln \left (x \right )+140816\right )}{\left (5+\ln \left (x \right )\right )^{4}}\) | \(84\) |
| norman | \(\frac {\left (500 \ln \left (2\right ) {\mathrm e}+1000 \ln \left (2\right )\right ) \ln \left (x \right )^{3}+\left (150 \,{\mathrm e}^{2} \ln \left (2\right )+8100 \ln \left (2\right ) {\mathrm e}+15600 \ln \left (2\right )\right ) \ln \left (x \right )^{2}+\left (20 \ln \left (2\right ) {\mathrm e}^{3}+1620 \,{\mathrm e}^{2} \ln \left (2\right )+43740 \ln \left (2\right ) {\mathrm e}+81160 \ln \left (2\right )\right ) \ln \left (x \right )+78732 \ln \left (2\right ) {\mathrm e}+140816 \ln \left (2\right )+4374 \,{\mathrm e}^{2} \ln \left (2\right )+108 \ln \left (2\right ) {\mathrm e}^{3}+{\mathrm e}^{4} \ln \left (2\right )}{\left (5+\ln \left (x \right )\right )^{4}}\) | \(112\) |
| parallelrisch | \(\frac {4374 \,{\mathrm e}^{2} \ln \left (2\right )+81160 \ln \left (2\right ) \ln \left (x \right )+108 \ln \left (2\right ) {\mathrm e}^{3}+{\mathrm e}^{4} \ln \left (2\right )+15600 \ln \left (x \right )^{2} \ln \left (2\right )+1000 \ln \left (2\right ) \ln \left (x \right )^{3}+78732 \ln \left (2\right ) {\mathrm e}+140816 \ln \left (2\right )+20 \ln \left (2\right ) \ln \left (x \right ) {\mathrm e}^{3}+150 \ln \left (2\right ) \ln \left (x \right )^{2} {\mathrm e}^{2}+500 \ln \left (2\right ) \ln \left (x \right )^{3} {\mathrm e}+1620 \ln \left (2\right ) \ln \left (x \right ) {\mathrm e}^{2}+8100 \,{\mathrm e} \ln \left (2\right ) \ln \left (x \right )^{2}+43740 \,{\mathrm e} \ln \left (2\right ) \ln \left (x \right )}{\ln \left (x \right )^{4}+20 \ln \left (x \right )^{3}+150 \ln \left (x \right )^{2}+500 \ln \left (x \right )+625}\) | \(142\) |
Input:
int(((-500*exp(1)-1000)*ln(2)*ln(x)^3+(-300*exp(1)^2-8700*exp(1)-16200)*ln (2)*ln(x)^2+(-60*exp(1)^3-3360*exp(1)^2-50220*exp(1)-87480)*ln(2)*ln(x)+(- 4*exp(1)^4-332*exp(1)^3-9396*exp(1)^2-96228*exp(1)-157464)*ln(2))/(x*ln(x) ^5+25*x*ln(x)^4+250*x*ln(x)^3+1250*x*ln(x)^2+3125*x*ln(x)+3125*x),x,method =_RETURNVERBOSE)
Output:
-4*ln(2)*(exp(1)+2)*(-1/3*(15*exp(2)+60*exp(1)+60)/(5+ln(x))^3-1/4*(exp(3) +6*exp(2)+12*exp(1)+8)/(5+ln(x))^4-125/(5+ln(x))-1/2*(75*exp(1)+150)/(5+ln (x))^2)
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (18) = 36\).
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 5.35 \[ \int \frac {\left (-157464-96228 e-9396 e^2-332 e^3-4 e^4\right ) \log (2)+\left (-87480-50220 e-3360 e^2-60 e^3\right ) \log (2) \log (x)+\left (-16200-8700 e-300 e^2\right ) \log (2) \log ^2(x)+(-1000-500 e) \log (2) \log ^3(x)}{3125 x+3125 x \log (x)+1250 x \log ^2(x)+250 x \log ^3(x)+25 x \log ^4(x)+x \log ^5(x)} \, dx=\frac {500 \, {\left (e + 2\right )} \log \left (2\right ) \log \left (x\right )^{3} + 150 \, {\left (e^{2} + 54 \, e + 104\right )} \log \left (2\right ) \log \left (x\right )^{2} + 20 \, {\left (e^{3} + 81 \, e^{2} + 2187 \, e + 4058\right )} \log \left (2\right ) \log \left (x\right ) + {\left (e^{4} + 108 \, e^{3} + 4374 \, e^{2} + 78732 \, e + 140816\right )} \log \left (2\right )}{\log \left (x\right )^{4} + 20 \, \log \left (x\right )^{3} + 150 \, \log \left (x\right )^{2} + 500 \, \log \left (x\right ) + 625} \] Input:
integrate(((-500*exp(1)-1000)*log(2)*log(x)^3+(-300*exp(1)^2-8700*exp(1)-1 6200)*log(2)*log(x)^2+(-60*exp(1)^3-3360*exp(1)^2-50220*exp(1)-87480)*log( 2)*log(x)+(-4*exp(1)^4-332*exp(1)^3-9396*exp(1)^2-96228*exp(1)-157464)*log (2))/(x*log(x)^5+25*x*log(x)^4+250*x*log(x)^3+1250*x*log(x)^2+3125*x*log(x )+3125*x),x, algorithm="fricas")
Output:
(500*(e + 2)*log(2)*log(x)^3 + 150*(e^2 + 54*e + 104)*log(2)*log(x)^2 + 20 *(e^3 + 81*e^2 + 2187*e + 4058)*log(2)*log(x) + (e^4 + 108*e^3 + 4374*e^2 + 78732*e + 140816)*log(2))/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*l og(x) + 625)
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (15) = 30\).
Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 8.29 \[ \int \frac {\left (-157464-96228 e-9396 e^2-332 e^3-4 e^4\right ) \log (2)+\left (-87480-50220 e-3360 e^2-60 e^3\right ) \log (2) \log (x)+\left (-16200-8700 e-300 e^2\right ) \log (2) \log ^2(x)+(-1000-500 e) \log (2) \log ^3(x)}{3125 x+3125 x \log (x)+1250 x \log ^2(x)+250 x \log ^3(x)+25 x \log ^4(x)+x \log ^5(x)} \, dx=\frac {\left (1000 \log {\left (2 \right )} + 500 e \log {\left (2 \right )}\right ) \log {\left (x \right )}^{3} + \left (150 e^{2} \log {\left (2 \right )} + 15600 \log {\left (2 \right )} + 8100 e \log {\left (2 \right )}\right ) \log {\left (x \right )}^{2} + \left (20 e^{3} \log {\left (2 \right )} + 1620 e^{2} \log {\left (2 \right )} + 81160 \log {\left (2 \right )} + 43740 e \log {\left (2 \right )}\right ) \log {\left (x \right )} + e^{4} \log {\left (2 \right )} + 108 e^{3} \log {\left (2 \right )} + 4374 e^{2} \log {\left (2 \right )} + 140816 \log {\left (2 \right )} + 78732 e \log {\left (2 \right )}}{\log {\left (x \right )}^{4} + 20 \log {\left (x \right )}^{3} + 150 \log {\left (x \right )}^{2} + 500 \log {\left (x \right )} + 625} \] Input:
integrate(((-500*exp(1)-1000)*ln(2)*ln(x)**3+(-300*exp(1)**2-8700*exp(1)-1 6200)*ln(2)*ln(x)**2+(-60*exp(1)**3-3360*exp(1)**2-50220*exp(1)-87480)*ln( 2)*ln(x)+(-4*exp(1)**4-332*exp(1)**3-9396*exp(1)**2-96228*exp(1)-157464)*l n(2))/(x*ln(x)**5+25*x*ln(x)**4+250*x*ln(x)**3+1250*x*ln(x)**2+3125*x*ln(x )+3125*x),x)
Output:
((1000*log(2) + 500*E*log(2))*log(x)**3 + (150*exp(2)*log(2) + 15600*log(2 ) + 8100*E*log(2))*log(x)**2 + (20*exp(3)*log(2) + 1620*exp(2)*log(2) + 81 160*log(2) + 43740*E*log(2))*log(x) + exp(4)*log(2) + 108*exp(3)*log(2) + 4374*exp(2)*log(2) + 140816*log(2) + 78732*E*log(2))/(log(x)**4 + 20*log(x )**3 + 150*log(x)**2 + 500*log(x) + 625)
Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (18) = 36\).
Time = 0.06 (sec) , antiderivative size = 692, normalized size of antiderivative = 40.71 \[ \int \frac {\left (-157464-96228 e-9396 e^2-332 e^3-4 e^4\right ) \log (2)+\left (-87480-50220 e-3360 e^2-60 e^3\right ) \log (2) \log (x)+\left (-16200-8700 e-300 e^2\right ) \log (2) \log ^2(x)+(-1000-500 e) \log (2) \log ^3(x)}{3125 x+3125 x \log (x)+1250 x \log ^2(x)+250 x \log ^3(x)+25 x \log ^4(x)+x \log ^5(x)} \, dx=\text {Too large to display} \] Input:
integrate(((-500*exp(1)-1000)*log(2)*log(x)^3+(-300*exp(1)^2-8700*exp(1)-1 6200)*log(2)*log(x)^2+(-60*exp(1)^3-3360*exp(1)^2-50220*exp(1)-87480)*log( 2)*log(x)+(-4*exp(1)^4-332*exp(1)^3-9396*exp(1)^2-96228*exp(1)-157464)*log (2))/(x*log(x)^5+25*x*log(x)^4+250*x*log(x)^3+1250*x*log(x)^2+3125*x*log(x )+3125*x),x, algorithm="maxima")
Output:
125*e*log(2)*log(x)^3/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 75*e^2*log(2)*log(x)^2/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 5 00*log(x) + 625) + 2175*e*log(2)*log(x)^2/(log(x)^4 + 20*log(x)^3 + 150*lo g(x)^2 + 500*log(x) + 625) + 250*log(2)*log(x)^3/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 25*(3*log(x) + 5)*e^2*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 125*(3*log(x)^2 + 15*log(x) + 25)*e*lo g(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 725*(3*log(x) + 5)*e*log (2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 15*e^3*log(2)*log(x)/(log (x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 840*e^2*log(2)*lo g(x)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 12555*e* log(2)*log(x)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 4050*log(2)*log(x)^2/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 250*(3*log(x)^2 + 15*log(x) + 25)*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 1350*(3*log(x) + 5)*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + e^4*log(2)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500 *log(x) + 625) + 83*e^3*log(2)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 50 0*log(x) + 625) + 5*e^3*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 2349*e^2*log(2)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 62 5) + 280*e^2*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 24057*e*l og(2)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 4185...
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (18) = 36\).
Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 6.53 \[ \int \frac {\left (-157464-96228 e-9396 e^2-332 e^3-4 e^4\right ) \log (2)+\left (-87480-50220 e-3360 e^2-60 e^3\right ) \log (2) \log (x)+\left (-16200-8700 e-300 e^2\right ) \log (2) \log ^2(x)+(-1000-500 e) \log (2) \log ^3(x)}{3125 x+3125 x \log (x)+1250 x \log ^2(x)+250 x \log ^3(x)+25 x \log ^4(x)+x \log ^5(x)} \, dx=\frac {500 \, e \log \left (2\right ) \log \left (x\right )^{3} + 150 \, e^{2} \log \left (2\right ) \log \left (x\right )^{2} + 8100 \, e \log \left (2\right ) \log \left (x\right )^{2} + 1000 \, \log \left (2\right ) \log \left (x\right )^{3} + 20 \, e^{3} \log \left (2\right ) \log \left (x\right ) + 1620 \, e^{2} \log \left (2\right ) \log \left (x\right ) + 43740 \, e \log \left (2\right ) \log \left (x\right ) + 15600 \, \log \left (2\right ) \log \left (x\right )^{2} + e^{4} \log \left (2\right ) + 108 \, e^{3} \log \left (2\right ) + 4374 \, e^{2} \log \left (2\right ) + 78732 \, e \log \left (2\right ) + 81160 \, \log \left (2\right ) \log \left (x\right ) + 140816 \, \log \left (2\right )}{{\left (\log \left (x\right ) + 5\right )}^{4}} \] Input:
integrate(((-500*exp(1)-1000)*log(2)*log(x)^3+(-300*exp(1)^2-8700*exp(1)-1 6200)*log(2)*log(x)^2+(-60*exp(1)^3-3360*exp(1)^2-50220*exp(1)-87480)*log( 2)*log(x)+(-4*exp(1)^4-332*exp(1)^3-9396*exp(1)^2-96228*exp(1)-157464)*log (2))/(x*log(x)^5+25*x*log(x)^4+250*x*log(x)^3+1250*x*log(x)^2+3125*x*log(x )+3125*x),x, algorithm="giac")
Output:
(500*e*log(2)*log(x)^3 + 150*e^2*log(2)*log(x)^2 + 8100*e*log(2)*log(x)^2 + 1000*log(2)*log(x)^3 + 20*e^3*log(2)*log(x) + 1620*e^2*log(2)*log(x) + 4 3740*e*log(2)*log(x) + 15600*log(2)*log(x)^2 + e^4*log(2) + 108*e^3*log(2) + 4374*e^2*log(2) + 78732*e*log(2) + 81160*log(2)*log(x) + 140816*log(2)) /(log(x) + 5)^4
Time = 3.93 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.65 \[ \int \frac {\left (-157464-96228 e-9396 e^2-332 e^3-4 e^4\right ) \log (2)+\left (-87480-50220 e-3360 e^2-60 e^3\right ) \log (2) \log (x)+\left (-16200-8700 e-300 e^2\right ) \log (2) \log ^2(x)+(-1000-500 e) \log (2) \log ^3(x)}{3125 x+3125 x \log (x)+1250 x \log ^2(x)+250 x \log ^3(x)+25 x \log ^4(x)+x \log ^5(x)} \, dx=\frac {500\,\ln \left (2\right )\,\left (\mathrm {e}+2\right )}{\ln \left (x\right )+5}+\frac {150\,\ln \left (2\right )\,{\left (\mathrm {e}+2\right )}^2}{{\left (\ln \left (x\right )+5\right )}^2}+\frac {20\,\ln \left (2\right )\,{\left (\mathrm {e}+2\right )}^3}{{\left (\ln \left (x\right )+5\right )}^3}+\frac {\ln \left (2\right )\,{\left (\mathrm {e}+2\right )}^4}{{\left (\ln \left (x\right )+5\right )}^4} \] Input:
int(-(log(2)*(96228*exp(1) + 9396*exp(2) + 332*exp(3) + 4*exp(4) + 157464) + log(2)*log(x)^3*(500*exp(1) + 1000) + log(2)*log(x)*(50220*exp(1) + 336 0*exp(2) + 60*exp(3) + 87480) + log(2)*log(x)^2*(8700*exp(1) + 300*exp(2) + 16200))/(3125*x + 1250*x*log(x)^2 + 250*x*log(x)^3 + 25*x*log(x)^4 + x*l og(x)^5 + 3125*x*log(x)),x)
Output:
(500*log(2)*(exp(1) + 2))/(log(x) + 5) + (150*log(2)*(exp(1) + 2)^2)/(log( x) + 5)^2 + (20*log(2)*(exp(1) + 2)^3)/(log(x) + 5)^3 + (log(2)*(exp(1) + 2)^4)/(log(x) + 5)^4
Time = 0.18 (sec) , antiderivative size = 103, normalized size of antiderivative = 6.06 \[ \int \frac {\left (-157464-96228 e-9396 e^2-332 e^3-4 e^4\right ) \log (2)+\left (-87480-50220 e-3360 e^2-60 e^3\right ) \log (2) \log (x)+\left (-16200-8700 e-300 e^2\right ) \log (2) \log ^2(x)+(-1000-500 e) \log (2) \log ^3(x)}{3125 x+3125 x \log (x)+1250 x \log ^2(x)+250 x \log ^3(x)+25 x \log ^4(x)+x \log ^5(x)} \, dx=\frac {\mathrm {log}\left (2\right ) \left (-25 \mathrm {log}\left (x \right )^{4} e -50 \mathrm {log}\left (x \right )^{4}+150 \mathrm {log}\left (x \right )^{2} e^{2}+4350 \mathrm {log}\left (x \right )^{2} e +8100 \mathrm {log}\left (x \right )^{2}+20 \,\mathrm {log}\left (x \right ) e^{3}+1620 \,\mathrm {log}\left (x \right ) e^{2}+31240 \,\mathrm {log}\left (x \right ) e +56160 \,\mathrm {log}\left (x \right )+e^{4}+108 e^{3}+4374 e^{2}+63107 e +109566\right )}{\mathrm {log}\left (x \right )^{4}+20 \mathrm {log}\left (x \right )^{3}+150 \mathrm {log}\left (x \right )^{2}+500 \,\mathrm {log}\left (x \right )+625} \] Input:
int(((-500*exp(1)-1000)*log(2)*log(x)^3+(-300*exp(1)^2-8700*exp(1)-16200)* log(2)*log(x)^2+(-60*exp(1)^3-3360*exp(1)^2-50220*exp(1)-87480)*log(2)*log (x)+(-4*exp(1)^4-332*exp(1)^3-9396*exp(1)^2-96228*exp(1)-157464)*log(2))/( x*log(x)^5+25*x*log(x)^4+250*x*log(x)^3+1250*x*log(x)^2+3125*x*log(x)+3125 *x),x)
Output:
(log(2)*( - 25*log(x)**4*e - 50*log(x)**4 + 150*log(x)**2*e**2 + 4350*log( x)**2*e + 8100*log(x)**2 + 20*log(x)*e**3 + 1620*log(x)*e**2 + 31240*log(x )*e + 56160*log(x) + e**4 + 108*e**3 + 4374*e**2 + 63107*e + 109566))/(log (x)**4 + 20*log(x)**3 + 150*log(x)**2 + 500*log(x) + 625)