\(\int \frac {625+(13500+7500 x) \log (2)+(97200+108000 x+37200 x^2) \log ^2(2)+(233280+388800 x+216000 x^2+68800 x^3) \log ^3(2)}{125 x^2+(2700 x^2+1500 x^3) \log (2)+(19440 x^2+21600 x^3+6000 x^4) \log ^2(2)+(46656 x^2+77760 x^3+43200 x^4+8000 x^5) \log ^3(2)} \, dx\) [864]

Optimal result

Integrand size = 115, antiderivative size = 31 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=-\frac {5}{x}+\frac {x^2}{\left (x+\frac {9}{5+\frac {5}{4 x \log (2)}}\right )^2} \] Output:

x^2/(9/(5/4/x/ln(2)+5)+x)^2-5/x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=5 \left (-\frac {1}{x}-\frac {72 \log (2) (5+18 \log (2)+20 x \log (2))}{5 (5+36 \log (2)+20 x \log (2))^2}\right ) \] Input:

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]
 

Output:

5*(-x^(-1) - (72*Log[2]*(5 + 18*Log[2] + 20*x*Log[2]))/(5*(5 + 36*Log[2] + 
 20*x*Log[2])^2))
 

Rubi [A] (verified)

Time = 0.43 (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

\(\displaystyle -\frac {5}{x}+\frac {1296 \log ^2(2)}{(20 x \log (2)+5+36 \log (2))^2}-\frac {72 \log (2)}{20 x \log (2)+5+36 \log (2)}\)

Input:

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]
 

Output:

-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])
 

Defintions of rubi rules used

rule 2007

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

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])
 

rule 2123

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])
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39

method	result	size
default	\(-\frac {5}{x}-\frac {72 \ln \left (2\right )}{20 x \ln \left (2\right )+36 \ln \left (2\right )+5}+\frac {1296 \ln \left (2\right )^{2}}{\left (20 x \ln \left (2\right )+36 \ln \left (2\right )+5\right )^{2}}\)	\(43\)
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	\(-\frac {1376000 x^{2} \ln \left (2\right )^{4}+3398400 x \ln \left (2\right )^{4}+2592000 \ln \left (2\right )^{4}+544000 x \ln \left (2\right )^{3}+720000 \ln \left (2\right )^{3}+50000 \ln \left (2\right )^{2}}{400 \ln \left (2\right )^{2} x \left (400 x^{2} \ln \left (2\right )^{2}+1440 x \ln \left (2\right )^{2}+1296 \ln \left (2\right )^{2}+200 x \ln \left (2\right )+360 \ln \left (2\right )+25\right )}\)	\(87\)

Input:

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)
 

Output:

-5/x-72*ln(2)/(20*x*ln(2)+36*ln(2)+5)+1296*ln(2)^2/(20*x*ln(2)+36*ln(2)+5) 
^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (31) = 62\).

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=-\frac {16 \, {\left (215 \, x^{2} + 531 \, x + 405\right )} \log \left (2\right )^{2} + 40 \, {\left (34 \, x + 45\right )} \log \left (2\right ) + 125}{16 \, {\left (25 \, x^{3} + 90 \, x^{2} + 81 \, x\right )} \log \left (2\right )^{2} + 40 \, {\left (5 \, x^{2} + 9 \, x\right )} \log \left (2\right ) + 25 \, x} \] Input:

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="fricas")
 

Output:

-(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)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (20) = 40\).

Time = 0.97 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.52 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=\frac {- 3440 x^{2} \log {\left (2 \right )}^{2} + x \left (- 8496 \log {\left (2 \right )}^{2} - 1360 \log {\left (2 \right )}\right ) - 6480 \log {\left (2 \right )}^{2} - 1800 \log {\left (2 \right )} - 125}{400 x^{3} \log {\left (2 \right )}^{2} + x^{2} \cdot \left (200 \log {\left (2 \right )} + 1440 \log {\left (2 \right )}^{2}\right ) + x \left (25 + 360 \log {\left (2 \right )} + 1296 \log {\left (2 \right )}^{2}\right )} \] Input:

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)
 

Output:

(-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))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (31) = 62\).

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=-\frac {3440 \, x^{2} \log \left (2\right )^{2} + 16 \, {\left (531 \, \log \left (2\right )^{2} + 85 \, \log \left (2\right )\right )} x + 6480 \, \log \left (2\right )^{2} + 1800 \, \log \left (2\right ) + 125}{400 \, x^{3} \log \left (2\right )^{2} + 40 \, {\left (36 \, \log \left (2\right )^{2} + 5 \, \log \left (2\right )\right )} x^{2} + {\left (1296 \, \log \left (2\right )^{2} + 360 \, \log \left (2\right ) + 25\right )} x} \] Input:

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="maxima")
 

Output:

-(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)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=-\frac {72 \, {\left (20 \, x \log \left (2\right )^{2} + 18 \, \log \left (2\right )^{2} + 5 \, \log \left (2\right )\right )}}{{\left (20 \, x \log \left (2\right ) + 36 \, \log \left (2\right ) + 5\right )}^{2}} - \frac {5}{x} \] Input:

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="giac")
 

Output:

-72*(20*x*log(2)^2 + 18*log(2)^2 + 5*log(2))/(20*x*log(2) + 36*log(2) + 5) 
^2 - 5/x
 

Mupad [B] (verification not implemented)

Time = 2.77 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=\frac {1296\,{\ln \left (2\right )}^2}{{\left (36\,\ln \left (2\right )+20\,x\,\ln \left (2\right )+5\right )}^2}-\frac {5}{x}-\frac {72\,\ln \left (2\right )}{36\,\ln \left (2\right )+20\,x\,\ln \left (2\right )+5} \] Input:

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)
 

Output:

(1296*log(2)^2)/(36*log(2) + 20*x*log(2) + 5)^2 - 5/x - (72*log(2))/(36*lo 
g(2) + 20*x*log(2) + 5)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.48 \[ \int \frac {625+(13500+7500 x) \log (2)+\left (97200+108000 x+37200 x^2\right ) \log ^2(2)+\left (233280+388800 x+216000 x^2+68800 x^3\right ) \log ^3(2)}{125 x^2+\left (2700 x^2+1500 x^3\right ) \log (2)+\left (19440 x^2+21600 x^3+6000 x^4\right ) \log ^2(2)+\left (46656 x^2+77760 x^3+43200 x^4+8000 x^5\right ) \log ^3(2)} \, dx=\frac {34400 \mathrm {log}\left (2\right )^{3} x^{3}-194400 \mathrm {log}\left (2\right )^{3} x -233280 \mathrm {log}\left (2\right )^{3}-60480 \mathrm {log}\left (2\right )^{2} x -97200 \mathrm {log}\left (2\right )^{2}-4650 \,\mathrm {log}\left (2\right ) x -13500 \,\mathrm {log}\left (2\right )-625}{x \left (14400 \mathrm {log}\left (2\right )^{3} x^{2}+51840 \mathrm {log}\left (2\right )^{3} x +46656 \mathrm {log}\left (2\right )^{3}+2000 \mathrm {log}\left (2\right )^{2} x^{2}+14400 \mathrm {log}\left (2\right )^{2} x +19440 \mathrm {log}\left (2\right )^{2}+1000 \,\mathrm {log}\left (2\right ) x +2700 \,\mathrm {log}\left (2\right )+125\right )} \] Input:

int(((68800*x^3+216000*x^2+388800*x+233280)*log(2)^3+(37200*x^2+108000*x+9 
7200)*log(2)^2+(7500*x+13500)*log(2)+625)/((8000*x^5+43200*x^4+77760*x^3+4 
6656*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)
 

Output:

(5*(6880*log(2)**3*x**3 - 38880*log(2)**3*x - 46656*log(2)**3 - 12096*log( 
2)**2*x - 19440*log(2)**2 - 930*log(2)*x - 2700*log(2) - 125))/(x*(14400*l 
og(2)**3*x**2 + 51840*log(2)**3*x + 46656*log(2)**3 + 2000*log(2)**2*x**2 
+ 14400*log(2)**2*x + 19440*log(2)**2 + 1000*log(2)*x + 2700*log(2) + 125) 
)