\(\int \frac {-202500-202500 x-67500 x^2-7500 x^3+(50625+50625 x+16875 x^2+1875 x^3) \log (3 x)+(40500+35100 x+7200 x^2-900 x^3-300 x^4) \log ^2(3 x)+(-20250-14850 x+1800 x^3+300 x^4) \log ^3(3 x)+(2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5) \log ^5(3 x)}{(27+27 x+9 x^2+x^3) \log ^5(3 x)} \, dx\) [2351]

Optimal result

Integrand size = 138, antiderivative size = 23 \[ \int \frac {-202500-202500 x-67500 x^2-7500 x^3+\left (50625+50625 x+16875 x^2+1875 x^3\right ) \log (3 x)+\left (40500+35100 x+7200 x^2-900 x^3-300 x^4\right ) \log ^2(3 x)+\left (-20250-14850 x+1800 x^3+300 x^4\right ) \log ^3(3 x)+\left (2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5\right ) \log ^5(3 x)}{\left (27+27 x+9 x^2+x^3\right ) \log ^5(3 x)} \, dx=3 x \left (-6+x+\frac {3}{3+x}+\frac {25}{\log ^2(3 x)}\right )^2 \] Output:

3*x*(x-6+25/ln(3*x)^2+3/(3+x))^2
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(23)=46\).

Time = 1.51 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.57 \[ \int \frac {-202500-202500 x-67500 x^2-7500 x^3+\left (50625+50625 x+16875 x^2+1875 x^3\right ) \log (3 x)+\left (40500+35100 x+7200 x^2-900 x^3-300 x^4\right ) \log ^2(3 x)+\left (-20250-14850 x+1800 x^3+300 x^4\right ) \log ^3(3 x)+\left (2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5\right ) \log ^5(3 x)}{\left (27+27 x+9 x^2+x^3\right ) \log ^5(3 x)} \, dx=126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}+\frac {150 x \left (-15-3 x+x^2\right )}{(3+x) \log ^2(3 x)} \] Input:

Integrate[(-202500 - 202500*x - 67500*x^2 - 7500*x^3 + (50625 + 50625*x + 
16875*x^2 + 1875*x^3)*Log[3*x] + (40500 + 35100*x + 7200*x^2 - 900*x^3 - 3 
00*x^4)*Log[3*x]^2 + (-20250 - 14850*x + 1800*x^3 + 300*x^4)*Log[3*x]^3 + 
(2025 + 945*x - 567*x^2 - 279*x^3 + 9*x^4 + 9*x^5)*Log[3*x]^5)/((27 + 27*x 
 + 9*x^2 + x^3)*Log[3*x]^5),x]
 

Output:

126*x - 36*x^2 + 3*x^3 - 81/(3 + x)^2 + 513/(3 + x) + (1875*x)/Log[3*x]^4 
+ (150*x*(-15 - 3*x + x^2))/((3 + x)*Log[3*x]^2)
 

Rubi [F]

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.

\(\displaystyle \int \frac {-7500 x^3-67500 x^2+\left (300 x^4+1800 x^3-14850 x-20250\right ) \log ^3(3 x)+\left (1875 x^3+16875 x^2+50625 x+50625\right ) \log (3 x)+\left (-300 x^4-900 x^3+7200 x^2+35100 x+40500\right ) \log ^2(3 x)+\left (9 x^5+9 x^4-279 x^3-567 x^2+945 x+2025\right ) \log ^5(3 x)-202500 x-202500}{\left (x^3+9 x^2+27 x+27\right ) \log ^5(3 x)} \, dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {-7500 x^3-67500 x^2+\left (300 x^4+1800 x^3-14850 x-20250\right ) \log ^3(3 x)+\left (1875 x^3+16875 x^2+50625 x+50625\right ) \log (3 x)+\left (-300 x^4-900 x^3+7200 x^2+35100 x+40500\right ) \log ^2(3 x)+\left (9 x^5+9 x^4-279 x^3-567 x^2+945 x+2025\right ) \log ^5(3 x)-202500 x-202500}{(x+3)^3 \log ^5(3 x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {300 \left (x^2-3 x-15\right )}{(x+3) \log ^3(3 x)}+\frac {9 \left (x^3+4 x^2-4 x-15\right ) \left (x^2-3 x-15\right )}{(x+3)^3}+\frac {150 \left (2 x^3+6 x^2-18 x-45\right )}{(x+3)^2 \log ^2(3 x)}-\frac {7500}{\log ^5(3 x)}+\frac {1875}{\log ^4(3 x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -300 \int \frac {x^2-3 x-15}{(x+3) \log ^3(3 x)}dx+150 \int \frac {2 x^3+6 x^2-18 x-45}{(x+3)^2 \log ^2(3 x)}dx+\frac {3 x \left (-x^2+3 x+15\right )^2}{(x+3)^2}+\frac {1875 x}{\log ^4(3 x)}\)

Input:

Int[(-202500 - 202500*x - 67500*x^2 - 7500*x^3 + (50625 + 50625*x + 16875* 
x^2 + 1875*x^3)*Log[3*x] + (40500 + 35100*x + 7200*x^2 - 900*x^3 - 300*x^4 
)*Log[3*x]^2 + (-20250 - 14850*x + 1800*x^3 + 300*x^4)*Log[3*x]^3 + (2025 
+ 945*x - 567*x^2 - 279*x^3 + 9*x^4 + 9*x^5)*Log[3*x]^5)/((27 + 27*x + 9*x 
^2 + x^3)*Log[3*x]^5),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(82\) vs. \(2(23)=46\).

Time = 42.54 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.61

method	result	size
parts	\(\frac {1875 x}{\ln \left (3 x \right )^{4}}+3 x^{3}-36 x^{2}+126 x +\frac {513}{3+x}-\frac {81}{\left (3+x \right )^{2}}+\frac {150 x^{2}}{\left (\ln \left (3\right )+\ln \left (x \right )\right )^{2}}+\frac {450}{\left (\ln \left (3\right )+\ln \left (x \right )\right )^{2}}-\frac {900 x}{\left (\ln \left (3\right )+\ln \left (x \right )\right )^{2}}-\frac {1350}{\left (3+x \right ) \left (\ln \left (3\right )+\ln \left (x \right )\right )^{2}}\)	\(83\)
risch	\(\frac {3 x^{5}-18 x^{4}-63 x^{3}+432 x^{2}+1647 x +1458}{x^{2}+6 x +9}+\frac {75 x \left (2 \ln \left (3 x \right )^{2} x^{2}-6 \ln \left (3 x \right )^{2} x -30 \ln \left (3 x \right )^{2}+25 x +75\right )}{\left (3+x \right ) \ln \left (3 x \right )^{4}}\)	\(84\)
derivativedivides	\(\frac {1875 x}{\ln \left (3 x \right )^{4}}+3 x^{3}-36 x^{2}+126 x -\frac {729}{\left (3 x +9\right )^{2}}+\frac {1539}{3 x +9}+\frac {150 x^{2}}{\ln \left (3 x \right )^{2}}+\frac {450}{\ln \left (3 x \right )^{2}}-\frac {900 x}{\ln \left (3 x \right )^{2}}-\frac {4050}{\left (3 x +9\right ) \ln \left (3 x \right )^{2}}\)	\(85\)
default	\(\frac {1875 x}{\ln \left (3 x \right )^{4}}+3 x^{3}-36 x^{2}+126 x -\frac {729}{\left (3 x +9\right )^{2}}+\frac {1539}{3 x +9}+\frac {150 x^{2}}{\ln \left (3 x \right )^{2}}+\frac {450}{\ln \left (3 x \right )^{2}}-\frac {900 x}{\ln \left (3 x \right )^{2}}-\frac {4050}{\left (3 x +9\right ) \ln \left (3 x \right )^{2}}\)	\(85\)
parallelrisch	\(-\frac {-18 \ln \left (3 x \right )^{4} x^{5}+108 \ln \left (3 x \right )^{4} x^{4}+378 \ln \left (3 x \right )^{4} x^{3}-900 \ln \left (3 x \right )^{2} x^{4}-945 \ln \left (3 x \right )^{4} x^{2}+21600 \ln \left (3 x \right )^{2} x^{2}+6075 \ln \left (3 x \right )^{4}-11250 x^{3}+40500 \ln \left (3 x \right )^{2} x -67500 x^{2}-101250 x}{6 \ln \left (3 x \right )^{4} \left (x^{2}+6 x +9\right )}\)	\(116\)

Input:

int(((9*x^5+9*x^4-279*x^3-567*x^2+945*x+2025)*ln(3*x)^5+(300*x^4+1800*x^3- 
14850*x-20250)*ln(3*x)^3+(-300*x^4-900*x^3+7200*x^2+35100*x+40500)*ln(3*x) 
^2+(1875*x^3+16875*x^2+50625*x+50625)*ln(3*x)-7500*x^3-67500*x^2-202500*x- 
202500)/(x^3+9*x^2+27*x+27)/ln(3*x)^5,x,method=_RETURNVERBOSE)
 

Output:

1875*x/ln(3*x)^4+3*x^3-36*x^2+126*x+513/(3+x)-81/(3+x)^2+150*x^2/(ln(3)+ln 
(x))^2+450/(ln(3)+ln(x))^2-900*x/(ln(3)+ln(x))^2-1350/(3+x)/(ln(3)+ln(x))^ 
2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (23) = 46\).

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.57 \[ \int \frac {-202500-202500 x-67500 x^2-7500 x^3+\left (50625+50625 x+16875 x^2+1875 x^3\right ) \log (3 x)+\left (40500+35100 x+7200 x^2-900 x^3-300 x^4\right ) \log ^2(3 x)+\left (-20250-14850 x+1800 x^3+300 x^4\right ) \log ^3(3 x)+\left (2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5\right ) \log ^5(3 x)}{\left (27+27 x+9 x^2+x^3\right ) \log ^5(3 x)} \, dx=\frac {3 \, {\left ({\left (x^{5} - 6 \, x^{4} - 21 \, x^{3} + 144 \, x^{2} + 549 \, x + 486\right )} \log \left (3 \, x\right )^{4} + 625 \, x^{3} + 50 \, {\left (x^{4} - 24 \, x^{2} - 45 \, x\right )} \log \left (3 \, x\right )^{2} + 3750 \, x^{2} + 5625 \, x\right )}}{{\left (x^{2} + 6 \, x + 9\right )} \log \left (3 \, x\right )^{4}} \] Input:

integrate(((9*x^5+9*x^4-279*x^3-567*x^2+945*x+2025)*log(3*x)^5+(300*x^4+18 
00*x^3-14850*x-20250)*log(3*x)^3+(-300*x^4-900*x^3+7200*x^2+35100*x+40500) 
*log(3*x)^2+(1875*x^3+16875*x^2+50625*x+50625)*log(3*x)-7500*x^3-67500*x^2 
-202500*x-202500)/(x^3+9*x^2+27*x+27)/log(3*x)^5,x, algorithm="fricas")
 

Output:

3*((x^5 - 6*x^4 - 21*x^3 + 144*x^2 + 549*x + 486)*log(3*x)^4 + 625*x^3 + 5 
0*(x^4 - 24*x^2 - 45*x)*log(3*x)^2 + 3750*x^2 + 5625*x)/((x^2 + 6*x + 9)*l 
og(3*x)^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).

Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.83 \[ \int \frac {-202500-202500 x-67500 x^2-7500 x^3+\left (50625+50625 x+16875 x^2+1875 x^3\right ) \log (3 x)+\left (40500+35100 x+7200 x^2-900 x^3-300 x^4\right ) \log ^2(3 x)+\left (-20250-14850 x+1800 x^3+300 x^4\right ) \log ^3(3 x)+\left (2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5\right ) \log ^5(3 x)}{\left (27+27 x+9 x^2+x^3\right ) \log ^5(3 x)} \, dx=3 x^{3} - 36 x^{2} + 126 x + \frac {513 x + 1458}{x^{2} + 6 x + 9} + \frac {1875 x^{2} + 5625 x + \left (150 x^{3} - 450 x^{2} - 2250 x\right ) \log {\left (3 x \right )}^{2}}{\left (x + 3\right ) \log {\left (3 x \right )}^{4}} \] Input:

integrate(((9*x**5+9*x**4-279*x**3-567*x**2+945*x+2025)*ln(3*x)**5+(300*x* 
*4+1800*x**3-14850*x-20250)*ln(3*x)**3+(-300*x**4-900*x**3+7200*x**2+35100 
*x+40500)*ln(3*x)**2+(1875*x**3+16875*x**2+50625*x+50625)*ln(3*x)-7500*x** 
3-67500*x**2-202500*x-202500)/(x**3+9*x**2+27*x+27)/ln(3*x)**5,x)
 

Output:

3*x**3 - 36*x**2 + 126*x + (513*x + 1458)/(x**2 + 6*x + 9) + (1875*x**2 + 
5625*x + (150*x**3 - 450*x**2 - 2250*x)*log(3*x)**2)/((x + 3)*log(3*x)**4)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (23) = 46\).

Time = 0.16 (sec) , antiderivative size = 410, normalized size of antiderivative = 17.83 \[ \int \frac {-202500-202500 x-67500 x^2-7500 x^3+\left (50625+50625 x+16875 x^2+1875 x^3\right ) \log (3 x)+\left (40500+35100 x+7200 x^2-900 x^3-300 x^4\right ) \log ^2(3 x)+\left (-20250-14850 x+1800 x^3+300 x^4\right ) \log ^3(3 x)+\left (2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5\right ) \log ^5(3 x)}{\left (27+27 x+9 x^2+x^3\right ) \log ^5(3 x)} \, dx =\text {Too large to display} \] Input:

integrate(((9*x^5+9*x^4-279*x^3-567*x^2+945*x+2025)*log(3*x)^5+(300*x^4+18 
00*x^3-14850*x-20250)*log(3*x)^3+(-300*x^4-900*x^3+7200*x^2+35100*x+40500) 
*log(3*x)^2+(1875*x^3+16875*x^2+50625*x+50625)*log(3*x)-7500*x^3-67500*x^2 
-202500*x-202500)/(x^3+9*x^2+27*x+27)/log(3*x)^5,x, algorithm="maxima")
 

Output:

3*(x^5*log(3)^4 - 2*(3*log(3)^4 - 25*log(3)^2)*x^4 + (x^5 - 6*x^4 - 21*x^3 
 + 144*x^2 + 549*x + 486)*log(x)^4 - (21*log(3)^4 - 625)*x^3 + 486*log(3)^ 
4 + 4*(x^5*log(3) - 6*x^4*log(3) - 21*x^3*log(3) + 144*x^2*log(3) + 549*x* 
log(3) + 486*log(3))*log(x)^3 + 6*(24*log(3)^4 - 200*log(3)^2 + 625)*x^2 + 
 2*(3*x^5*log(3)^2 - (18*log(3)^2 - 25)*x^4 - 63*x^3*log(3)^2 + 24*(18*log 
(3)^2 - 25)*x^2 + 9*(183*log(3)^2 - 125)*x + 1458*log(3)^2)*log(x)^2 + 9*( 
61*log(3)^4 - 250*log(3)^2 + 625)*x + 4*(x^5*log(3)^3 - 21*x^3*log(3)^3 - 
(6*log(3)^3 - 25*log(3))*x^4 + 24*(6*log(3)^3 - 25*log(3))*x^2 + 486*log(3 
)^3 + 9*(61*log(3)^3 - 125*log(3))*x)*log(x))/(x^2*log(3)^4 + 6*x*log(3)^4 
 + (x^2 + 6*x + 9)*log(x)^4 + 9*log(3)^4 + 4*(x^2*log(3) + 6*x*log(3) + 9* 
log(3))*log(x)^3 + 6*(x^2*log(3)^2 + 6*x*log(3)^2 + 9*log(3)^2)*log(x)^2 + 
 4*(x^2*log(3)^3 + 6*x*log(3)^3 + 9*log(3)^3)*log(x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (23) = 46\).

Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00 \[ \int \frac {-202500-202500 x-67500 x^2-7500 x^3+\left (50625+50625 x+16875 x^2+1875 x^3\right ) \log (3 x)+\left (40500+35100 x+7200 x^2-900 x^3-300 x^4\right ) \log ^2(3 x)+\left (-20250-14850 x+1800 x^3+300 x^4\right ) \log ^3(3 x)+\left (2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5\right ) \log ^5(3 x)}{\left (27+27 x+9 x^2+x^3\right ) \log ^5(3 x)} \, dx=3 \, x^{3} - 36 \, x^{2} + 126 \, x + \frac {75 \, {\left (2 \, x^{3} \log \left (3 \, x\right )^{2} - 6 \, x^{2} \log \left (3 \, x\right )^{2} - 30 \, x \log \left (3 \, x\right )^{2} + 25 \, x^{2} + 75 \, x\right )}}{x \log \left (3 \, x\right )^{4} + 3 \, \log \left (3 \, x\right )^{4}} + \frac {27 \, {\left (19 \, x + 54\right )}}{x^{2} + 6 \, x + 9} \] Input:

integrate(((9*x^5+9*x^4-279*x^3-567*x^2+945*x+2025)*log(3*x)^5+(300*x^4+18 
00*x^3-14850*x-20250)*log(3*x)^3+(-300*x^4-900*x^3+7200*x^2+35100*x+40500) 
*log(3*x)^2+(1875*x^3+16875*x^2+50625*x+50625)*log(3*x)-7500*x^3-67500*x^2 
-202500*x-202500)/(x^3+9*x^2+27*x+27)/log(3*x)^5,x, algorithm="giac")
 

Output:

3*x^3 - 36*x^2 + 126*x + 75*(2*x^3*log(3*x)^2 - 6*x^2*log(3*x)^2 - 30*x*lo 
g(3*x)^2 + 25*x^2 + 75*x)/(x*log(3*x)^4 + 3*log(3*x)^4) + 27*(19*x + 54)/( 
x^2 + 6*x + 9)
 

Mupad [B] (verification not implemented)

Time = 2.89 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.96 \[ \int \frac {-202500-202500 x-67500 x^2-7500 x^3+\left (50625+50625 x+16875 x^2+1875 x^3\right ) \log (3 x)+\left (40500+35100 x+7200 x^2-900 x^3-300 x^4\right ) \log ^2(3 x)+\left (-20250-14850 x+1800 x^3+300 x^4\right ) \log ^3(3 x)+\left (2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5\right ) \log ^5(3 x)}{\left (27+27 x+9 x^2+x^3\right ) \log ^5(3 x)} \, dx=\frac {3\,x\,{\left (25\,x+75\right )}^2-6\,x\,{\ln \left (3\,x\right )}^2\,\left (25\,x+75\right )\,\left (-x^2+3\,x+15\right )}{{\ln \left (3\,x\right )}^4\,{\left (x+3\right )}^2}+\frac {3\,x\,{\left (-x^2+3\,x+15\right )}^2}{{\left (x+3\right )}^2} \] Input:

int(-(202500*x - log(3*x)^5*(945*x - 567*x^2 - 279*x^3 + 9*x^4 + 9*x^5 + 2 
025) - log(3*x)*(50625*x + 16875*x^2 + 1875*x^3 + 50625) + log(3*x)^3*(148 
50*x - 1800*x^3 - 300*x^4 + 20250) + 67500*x^2 + 7500*x^3 - log(3*x)^2*(35 
100*x + 7200*x^2 - 900*x^3 - 300*x^4 + 40500) + 202500)/(log(3*x)^5*(27*x 
+ 9*x^2 + x^3 + 27)),x)
 

Output:

(3*x*(25*x + 75)^2 - 6*x*log(3*x)^2*(25*x + 75)*(3*x - x^2 + 15))/(log(3*x 
)^4*(x + 3)^2) + (3*x*(3*x - x^2 + 15)^2)/(x + 3)^2
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 5.00 \[ \int \frac {-202500-202500 x-67500 x^2-7500 x^3+\left (50625+50625 x+16875 x^2+1875 x^3\right ) \log (3 x)+\left (40500+35100 x+7200 x^2-900 x^3-300 x^4\right ) \log ^2(3 x)+\left (-20250-14850 x+1800 x^3+300 x^4\right ) \log ^3(3 x)+\left (2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5\right ) \log ^5(3 x)}{\left (27+27 x+9 x^2+x^3\right ) \log ^5(3 x)} \, dx=\frac {3 \mathrm {log}\left (3 x \right )^{4} x^{5}-18 \mathrm {log}\left (3 x \right )^{4} x^{4}-63 \mathrm {log}\left (3 x \right )^{4} x^{3}+\frac {315 \mathrm {log}\left (3 x \right )^{4} x^{2}}{2}-\frac {2025 \mathrm {log}\left (3 x \right )^{4}}{2}+150 \mathrm {log}\left (3 x \right )^{2} x^{4}-3600 \mathrm {log}\left (3 x \right )^{2} x^{2}-6750 \mathrm {log}\left (3 x \right )^{2} x +1875 x^{3}+11250 x^{2}+16875 x}{\mathrm {log}\left (3 x \right )^{4} \left (x^{2}+6 x +9\right )} \] Input:

int(((9*x^5+9*x^4-279*x^3-567*x^2+945*x+2025)*log(3*x)^5+(300*x^4+1800*x^3 
-14850*x-20250)*log(3*x)^3+(-300*x^4-900*x^3+7200*x^2+35100*x+40500)*log(3 
*x)^2+(1875*x^3+16875*x^2+50625*x+50625)*log(3*x)-7500*x^3-67500*x^2-20250 
0*x-202500)/(x^3+9*x^2+27*x+27)/log(3*x)^5,x)
 

Output:

(3*(2*log(3*x)**4*x**5 - 12*log(3*x)**4*x**4 - 42*log(3*x)**4*x**3 + 105*l 
og(3*x)**4*x**2 - 675*log(3*x)**4 + 100*log(3*x)**2*x**4 - 2400*log(3*x)** 
2*x**2 - 4500*log(3*x)**2*x + 1250*x**3 + 7500*x**2 + 11250*x))/(2*log(3*x 
)**4*(x**2 + 6*x + 9))