Integrand size = 115, antiderivative size = 31 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=-\frac {5}{x}+\frac {x^2}{\left (x+\frac {9}{5+\frac {5}{4 x \log (2)}}\right )^2} \]
Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=5 \left (-\frac {1}{x}-\frac {72 \log (2) (5+18 \log (2)+20 x \log (2))}{5 (5+36 \log (2)+20 x \log (2))^2}\right ) \]
Integrate[(625 + (13500 + 7500*x)*Log[2] + (97200 + 108000*x + 37200*x^2)* Log[2]^2 + (233280 + 388800*x + 216000*x^2 + 68800*x^3)*Log[2]^3)/(125*x^2 + (2700*x^2 + 1500*x^3)*Log[2] + (19440*x^2 + 21600*x^3 + 6000*x^4)*Log[2 ]^2 + (46656*x^2 + 77760*x^3 + 43200*x^4 + 8000*x^5)*Log[2]^3),x]
Time = 0.39 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {2026, 2007, 2123, 2009}
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.
\(\displaystyle \int \frac {\left (37200 x^2+108000 x+97200\right ) \log ^2(2)+\left (68800 x^3+216000 x^2+388800 x+233280\right ) \log ^3(2)+(7500 x+13500) \log (2)+625}{125 x^2+\left (1500 x^3+2700 x^2\right ) \log (2)+\left (6000 x^4+21600 x^3+19440 x^2\right ) \log ^2(2)+\left (8000 x^5+43200 x^4+77760 x^3+46656 x^2\right ) \log ^3(2)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (37200 x^2+108000 x+97200\right ) \log ^2(2)+\left (68800 x^3+216000 x^2+388800 x+233280\right ) \log ^3(2)+(7500 x+13500) \log (2)+625}{x^2 \left (8000 x^3 \log ^3(2)+1200 x^2 \log ^2(2) (5+36 \log (2))+60 x \log (2) (5+36 \log (2))^2+(5+36 \log (2))^3\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {\left (37200 x^2+108000 x+97200\right ) \log ^2(2)+\left (68800 x^3+216000 x^2+388800 x+233280\right ) \log ^3(2)+(7500 x+13500) \log (2)+625}{x^2 (20 x \log (2)+5+36 \log (2))^3}dx\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \int \left (\frac {5}{x^2}-\frac {51840 \log ^3(2)}{(20 x \log (2)+5+36 \log (2))^3}+\frac {1440 \log ^2(2)}{(20 x \log (2)+5+36 \log (2))^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5}{x}+\frac {1296 \log ^2(2)}{(20 x \log (2)+5+36 \log (2))^2}-\frac {72 \log (2)}{20 x \log (2)+5+36 \log (2)}\) |
Int[(625 + (13500 + 7500*x)*Log[2] + (97200 + 108000*x + 37200*x^2)*Log[2] ^2 + (233280 + 388800*x + 216000*x^2 + 68800*x^3)*Log[2]^3)/(125*x^2 + (27 00*x^2 + 1500*x^3)*Log[2] + (19440*x^2 + 21600*x^3 + 6000*x^4)*Log[2]^2 + (46656*x^2 + 77760*x^3 + 43200*x^4 + 8000*x^5)*Log[2]^3),x]
-5/x + (1296*Log[2]^2)/(5 + 36*Log[2] + 20*x*Log[2])^2 - (72*Log[2])/(5 + 36*Log[2] + 20*x*Log[2])
3.9.64.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), 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)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 1.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39
method | result | size |
default | \(-\frac {72 \ln \left (2\right )}{20 x \ln \left (2\right )+36 \ln \left (2\right )+5}+\frac {1296 \ln \left (2\right )^{2}}{\left (20 x \ln \left (2\right )+36 \ln \left (2\right )+5\right )^{2}}-\frac {5}{x}\) | \(43\) |
norman | \(\frac {-3440 x^{2} \ln \left (2\right )^{2}-\frac {\left (3398400 \ln \left (2\right )^{4}+544000 \ln \left (2\right )^{3}\right ) x}{400 \ln \left (2\right )^{2}}-6480 \ln \left (2\right )^{2}-1800 \ln \left (2\right )-125}{x \left (20 x \ln \left (2\right )+36 \ln \left (2\right )+5\right )^{2}}\) | \(59\) |
gosper | \(-\frac {3440 x^{2} \ln \left (2\right )^{2}+8496 x \ln \left (2\right )^{2}+6480 \ln \left (2\right )^{2}+1360 x \ln \left (2\right )+1800 \ln \left (2\right )+125}{x \left (400 x^{2} \ln \left (2\right )^{2}+1440 x \ln \left (2\right )^{2}+1296 \ln \left (2\right )^{2}+200 x \ln \left (2\right )+360 \ln \left (2\right )+25\right )}\) | \(74\) |
risch | \(\frac {-3440 x^{2} \ln \left (2\right )^{2}+400 \left (-\frac {531 \ln \left (2\right )^{2}}{25}-\frac {17 \ln \left (2\right )}{5}\right ) x -6480 \ln \left (2\right )^{2}-1800 \ln \left (2\right )-125}{x \left (400 x^{2} \ln \left (2\right )^{2}+1440 x \ln \left (2\right )^{2}+1296 \ln \left (2\right )^{2}+200 x \ln \left (2\right )+360 \ln \left (2\right )+25\right )}\) | \(75\) |
parallelrisch | \(-\frac {1376000 x^{2} \ln \left (2\right )^{4}+3398400 x \ln \left (2\right )^{4}+2592000 \ln \left (2\right )^{4}+544000 x \ln \left (2\right )^{3}+720000 \ln \left (2\right )^{3}+50000 \ln \left (2\right )^{2}}{400 \ln \left (2\right )^{2} x \left (400 x^{2} \ln \left (2\right )^{2}+1440 x \ln \left (2\right )^{2}+1296 \ln \left (2\right )^{2}+200 x \ln \left (2\right )+360 \ln \left (2\right )+25\right )}\) | \(87\) |
int(((68800*x^3+216000*x^2+388800*x+233280)*ln(2)^3+(37200*x^2+108000*x+97 200)*ln(2)^2+(7500*x+13500)*ln(2)+625)/((8000*x^5+43200*x^4+77760*x^3+4665 6*x^2)*ln(2)^3+(6000*x^4+21600*x^3+19440*x^2)*ln(2)^2+(1500*x^3+2700*x^2)* ln(2)+125*x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (31) = 62\).
Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=-\frac {16 \, {\left (215 \, x^{2} + 531 \, x + 405\right )} \log \left (2\right )^{2} + 40 \, {\left (34 \, x + 45\right )} \log \left (2\right ) + 125}{16 \, {\left (25 \, x^{3} + 90 \, x^{2} + 81 \, x\right )} \log \left (2\right )^{2} + 40 \, {\left (5 \, x^{2} + 9 \, x\right )} \log \left (2\right ) + 25 \, x} \]
integrate(((68800*x^3+216000*x^2+388800*x+233280)*log(2)^3+(37200*x^2+1080 00*x+97200)*log(2)^2+(7500*x+13500)*log(2)+625)/((8000*x^5+43200*x^4+77760 *x^3+46656*x^2)*log(2)^3+(6000*x^4+21600*x^3+19440*x^2)*log(2)^2+(1500*x^3 +2700*x^2)*log(2)+125*x^2),x, algorithm=\
-(16*(215*x^2 + 531*x + 405)*log(2)^2 + 40*(34*x + 45)*log(2) + 125)/(16*( 25*x^3 + 90*x^2 + 81*x)*log(2)^2 + 40*(5*x^2 + 9*x)*log(2) + 25*x)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (20) = 40\).
Time = 0.99 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.52 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=\frac {- 3440 x^{2} \log {\left (2 \right )}^{2} + x \left (- 8496 \log {\left (2 \right )}^{2} - 1360 \log {\left (2 \right )}\right ) - 6480 \log {\left (2 \right )}^{2} - 1800 \log {\left (2 \right )} - 125}{400 x^{3} \log {\left (2 \right )}^{2} + x^{2} \cdot \left (200 \log {\left (2 \right )} + 1440 \log {\left (2 \right )}^{2}\right ) + x \left (25 + 360 \log {\left (2 \right )} + 1296 \log {\left (2 \right )}^{2}\right )} \]
integrate(((68800*x**3+216000*x**2+388800*x+233280)*ln(2)**3+(37200*x**2+1 08000*x+97200)*ln(2)**2+(7500*x+13500)*ln(2)+625)/((8000*x**5+43200*x**4+7 7760*x**3+46656*x**2)*ln(2)**3+(6000*x**4+21600*x**3+19440*x**2)*ln(2)**2+ (1500*x**3+2700*x**2)*ln(2)+125*x**2),x)
(-3440*x**2*log(2)**2 + x*(-8496*log(2)**2 - 1360*log(2)) - 6480*log(2)**2 - 1800*log(2) - 125)/(400*x**3*log(2)**2 + x**2*(200*log(2) + 1440*log(2) **2) + x*(25 + 360*log(2) + 1296*log(2)**2))
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (31) = 62\).
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=-\frac {3440 \, x^{2} \log \left (2\right )^{2} + 16 \, {\left (531 \, \log \left (2\right )^{2} + 85 \, \log \left (2\right )\right )} x + 6480 \, \log \left (2\right )^{2} + 1800 \, \log \left (2\right ) + 125}{400 \, x^{3} \log \left (2\right )^{2} + 40 \, {\left (36 \, \log \left (2\right )^{2} + 5 \, \log \left (2\right )\right )} x^{2} + {\left (1296 \, \log \left (2\right )^{2} + 360 \, \log \left (2\right ) + 25\right )} x} \]
integrate(((68800*x^3+216000*x^2+388800*x+233280)*log(2)^3+(37200*x^2+1080 00*x+97200)*log(2)^2+(7500*x+13500)*log(2)+625)/((8000*x^5+43200*x^4+77760 *x^3+46656*x^2)*log(2)^3+(6000*x^4+21600*x^3+19440*x^2)*log(2)^2+(1500*x^3 +2700*x^2)*log(2)+125*x^2),x, algorithm=\
-(3440*x^2*log(2)^2 + 16*(531*log(2)^2 + 85*log(2))*x + 6480*log(2)^2 + 18 00*log(2) + 125)/(400*x^3*log(2)^2 + 40*(36*log(2)^2 + 5*log(2))*x^2 + (12 96*log(2)^2 + 360*log(2) + 25)*x)
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=-\frac {72 \, {\left (20 \, x \log \left (2\right )^{2} + 18 \, \log \left (2\right )^{2} + 5 \, \log \left (2\right )\right )}}{{\left (20 \, x \log \left (2\right ) + 36 \, \log \left (2\right ) + 5\right )}^{2}} - \frac {5}{x} \]
integrate(((68800*x^3+216000*x^2+388800*x+233280)*log(2)^3+(37200*x^2+1080 00*x+97200)*log(2)^2+(7500*x+13500)*log(2)+625)/((8000*x^5+43200*x^4+77760 *x^3+46656*x^2)*log(2)^3+(6000*x^4+21600*x^3+19440*x^2)*log(2)^2+(1500*x^3 +2700*x^2)*log(2)+125*x^2),x, algorithm=\
Time = 9.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=\frac {1296\,{\ln \left (2\right )}^2}{{\left (36\,\ln \left (2\right )+20\,x\,\ln \left (2\right )+5\right )}^2}-\frac {5}{x}-\frac {72\,\ln \left (2\right )}{36\,\ln \left (2\right )+20\,x\,\ln \left (2\right )+5} \]
int((log(2)*(7500*x + 13500) + log(2)^2*(108000*x + 37200*x^2 + 97200) + l og(2)^3*(388800*x + 216000*x^2 + 68800*x^3 + 233280) + 625)/(log(2)^2*(194 40*x^2 + 21600*x^3 + 6000*x^4) + log(2)^3*(46656*x^2 + 77760*x^3 + 43200*x ^4 + 8000*x^5) + log(2)*(2700*x^2 + 1500*x^3) + 125*x^2),x)