3.1.72 \(\int \frac {-147+7308 x-10311 x^2+5733 x^3-1470 x^4+147 x^5+(-105 x+42 x^2) \log (\frac {x}{3})}{2160900 x-3231060 x^2+1928101 x^3-559090 x^4+75411 x^5-3430 x^6+49 x^7+(2940 x-2198 x^2+490 x^3-14 x^4) \log (\frac {x}{3})+x \log ^2(\frac {x}{3})} \, dx\) [72]

3.1.72.1 Optimal result

Integrand size = 114, antiderivative size = 30 \[ \int \frac {-147+7308 x-10311 x^2+5733 x^3-1470 x^4+147 x^5+\left (-105 x+42 x^2\right ) \log \left (\frac {x}{3}\right )}{2160900 x-3231060 x^2+1928101 x^3-559090 x^4+75411 x^5-3430 x^6+49 x^7+\left (2940 x-2198 x^2+490 x^3-14 x^4\right ) \log \left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right )} \, dx=\frac {1}{10-\frac {x}{3}-\frac {\log \left (\frac {x}{3}\right )}{21 (-7+(5-x) x)}} \]

output

1/(10-1/3*x-1/3*ln(1/3*x)/(7*x*(5-x)-49))
 

3.1.72.2 Mathematica [A] (verified)

Time = 5.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {-147+7308 x-10311 x^2+5733 x^3-1470 x^4+147 x^5+\left (-105 x+42 x^2\right ) \log \left (\frac {x}{3}\right )}{2160900 x-3231060 x^2+1928101 x^3-559090 x^4+75411 x^5-3430 x^6+49 x^7+\left (2940 x-2198 x^2+490 x^3-14 x^4\right ) \log \left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right )} \, dx=\frac {21 \left (7-5 x+x^2\right )}{1470-1099 x+245 x^2-7 x^3+\log \left (\frac {x}{3}\right )} \]

input

Integrate[(-147 + 7308*x - 10311*x^2 + 5733*x^3 - 1470*x^4 + 147*x^5 + (-1 
05*x + 42*x^2)*Log[x/3])/(2160900*x - 3231060*x^2 + 1928101*x^3 - 559090*x 
^4 + 75411*x^5 - 3430*x^6 + 49*x^7 + (2940*x - 2198*x^2 + 490*x^3 - 14*x^4 
)*Log[x/3] + x*Log[x/3]^2),x]
 

output

(21*(7 - 5*x + x^2))/(1470 - 1099*x + 245*x^2 - 7*x^3 + Log[x/3])
 

3.1.72.3 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.

\(\displaystyle \int \frac {147 x^5-1470 x^4+5733 x^3-10311 x^2+\left (42 x^2-105 x\right ) \log \left (\frac {x}{3}\right )+7308 x-147}{49 x^7-3430 x^6+75411 x^5-559090 x^4+1928101 x^3-3231060 x^2+\left (-14 x^4+490 x^3-2198 x^2+2940 x\right ) \log \left (\frac {x}{3}\right )+2160900 x+x \log ^2\left (\frac {x}{3}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {147 x^5-1470 x^4+5733 x^3-10311 x^2+\left (42 x^2-105 x\right ) \log \left (\frac {x}{3}\right )+7308 x-147}{x \left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {1470 x^3}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}+\frac {5733 x^2}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}+\frac {42 x \log \left (\frac {x}{3}\right )}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}-\frac {10311 x}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}-\frac {105 \log \left (\frac {x}{3}\right )}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}+\frac {7308}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}-\frac {147}{x \left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}+\frac {147 x^4}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 7308 \int \frac {1}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}dx-147 \int \frac {1}{x \left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}dx-10311 \int \frac {x}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}dx+5733 \int \frac {x^2}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}dx-1470 \int \frac {x^3}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}dx-105 \int \frac {\log \left (\frac {x}{3}\right )}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}dx+42 \int \frac {x \log \left (\frac {x}{3}\right )}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}dx+147 \int \frac {x^4}{\left (7 x^3-245 x^2+1099 x-\log (x)-1470 \left (1-\frac {\log (3)}{1470}\right )\right )^2}dx\)

input

Int[(-147 + 7308*x - 10311*x^2 + 5733*x^3 - 1470*x^4 + 147*x^5 + (-105*x + 
 42*x^2)*Log[x/3])/(2160900*x - 3231060*x^2 + 1928101*x^3 - 559090*x^4 + 7 
5411*x^5 - 3430*x^6 + 49*x^7 + (2940*x - 2198*x^2 + 490*x^3 - 14*x^4)*Log[ 
x/3] + x*Log[x/3]^2),x]
 

output

$Aborted
 

3.1.72.3.1 Defintions of rubi rules used

rule 2009

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

rule 7292

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

rule 7293

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

3.1.72.4 Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {21 x^{2}-105 x +147}{-7 x^{3}+245 x^{2}+\ln \left (\frac {x}{3}\right )-1099 x +1470}\)	\(32\)
default	\(\frac {21 x^{2}-105 x +147}{-7 x^{3}+245 x^{2}+\ln \left (\frac {x}{3}\right )-1099 x +1470}\)	\(32\)
risch	\(-\frac {21 \left (x^{2}-5 x +7\right )}{7 x^{3}-245 x^{2}+1099 x -\ln \left (\frac {x}{3}\right )-1470}\)	\(34\)
norman	\(\frac {-21 x^{2}+105 x -147}{7 x^{3}-245 x^{2}+1099 x -\ln \left (\frac {x}{3}\right )-1470}\)	\(35\)
parallelrisch	\(\frac {-21 x^{2}+105 x -147}{7 x^{3}-245 x^{2}+1099 x -\ln \left (\frac {x}{3}\right )-1470}\)	\(35\)

input

int(((42*x^2-105*x)*ln(1/3*x)+147*x^5-1470*x^4+5733*x^3-10311*x^2+7308*x-1 
47)/(x*ln(1/3*x)^2+(-14*x^4+490*x^3-2198*x^2+2940*x)*ln(1/3*x)+49*x^7-3430 
*x^6+75411*x^5-559090*x^4+1928101*x^3-3231060*x^2+2160900*x),x,method=_RET 
URNVERBOSE)
 

output

21*(x^2-5*x+7)/(-7*x^3+245*x^2+ln(1/3*x)-1099*x+1470)
 

3.1.72.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {-147+7308 x-10311 x^2+5733 x^3-1470 x^4+147 x^5+\left (-105 x+42 x^2\right ) \log \left (\frac {x}{3}\right )}{2160900 x-3231060 x^2+1928101 x^3-559090 x^4+75411 x^5-3430 x^6+49 x^7+\left (2940 x-2198 x^2+490 x^3-14 x^4\right ) \log \left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right )} \, dx=-\frac {21 \, {\left (x^{2} - 5 \, x + 7\right )}}{7 \, x^{3} - 245 \, x^{2} + 1099 \, x - \log \left (\frac {1}{3} \, x\right ) - 1470} \]

input

integrate(((42*x^2-105*x)*log(1/3*x)+147*x^5-1470*x^4+5733*x^3-10311*x^2+7 
308*x-147)/(x*log(1/3*x)^2+(-14*x^4+490*x^3-2198*x^2+2940*x)*log(1/3*x)+49 
*x^7-3430*x^6+75411*x^5-559090*x^4+1928101*x^3-3231060*x^2+2160900*x),x, a 
lgorithm=\
 

output

-21*(x^2 - 5*x + 7)/(7*x^3 - 245*x^2 + 1099*x - log(1/3*x) - 1470)
 

3.1.72.6 Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-147+7308 x-10311 x^2+5733 x^3-1470 x^4+147 x^5+\left (-105 x+42 x^2\right ) \log \left (\frac {x}{3}\right )}{2160900 x-3231060 x^2+1928101 x^3-559090 x^4+75411 x^5-3430 x^6+49 x^7+\left (2940 x-2198 x^2+490 x^3-14 x^4\right ) \log \left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right )} \, dx=\frac {21 x^{2} - 105 x + 147}{- 7 x^{3} + 245 x^{2} - 1099 x + \log {\left (\frac {x}{3} \right )} + 1470} \]

input

integrate(((42*x**2-105*x)*ln(1/3*x)+147*x**5-1470*x**4+5733*x**3-10311*x* 
*2+7308*x-147)/(x*ln(1/3*x)**2+(-14*x**4+490*x**3-2198*x**2+2940*x)*ln(1/3 
*x)+49*x**7-3430*x**6+75411*x**5-559090*x**4+1928101*x**3-3231060*x**2+216 
0900*x),x)
 

output

(21*x**2 - 105*x + 147)/(-7*x**3 + 245*x**2 - 1099*x + log(x/3) + 1470)
 

3.1.72.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {-147+7308 x-10311 x^2+5733 x^3-1470 x^4+147 x^5+\left (-105 x+42 x^2\right ) \log \left (\frac {x}{3}\right )}{2160900 x-3231060 x^2+1928101 x^3-559090 x^4+75411 x^5-3430 x^6+49 x^7+\left (2940 x-2198 x^2+490 x^3-14 x^4\right ) \log \left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right )} \, dx=-\frac {21 \, {\left (x^{2} - 5 \, x + 7\right )}}{7 \, x^{3} - 245 \, x^{2} + 1099 \, x + \log \left (3\right ) - \log \left (x\right ) - 1470} \]

input

integrate(((42*x^2-105*x)*log(1/3*x)+147*x^5-1470*x^4+5733*x^3-10311*x^2+7 
308*x-147)/(x*log(1/3*x)^2+(-14*x^4+490*x^3-2198*x^2+2940*x)*log(1/3*x)+49 
*x^7-3430*x^6+75411*x^5-559090*x^4+1928101*x^3-3231060*x^2+2160900*x),x, a 
lgorithm=\
 

output

-21*(x^2 - 5*x + 7)/(7*x^3 - 245*x^2 + 1099*x + log(3) - log(x) - 1470)
 

3.1.72.8 Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {-147+7308 x-10311 x^2+5733 x^3-1470 x^4+147 x^5+\left (-105 x+42 x^2\right ) \log \left (\frac {x}{3}\right )}{2160900 x-3231060 x^2+1928101 x^3-559090 x^4+75411 x^5-3430 x^6+49 x^7+\left (2940 x-2198 x^2+490 x^3-14 x^4\right ) \log \left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right )} \, dx=-\frac {21 \, {\left (x^{2} - 5 \, x + 7\right )}}{7 \, x^{3} - 245 \, x^{2} + 1099 \, x - \log \left (\frac {1}{3} \, x\right ) - 1470} \]

input

integrate(((42*x^2-105*x)*log(1/3*x)+147*x^5-1470*x^4+5733*x^3-10311*x^2+7 
308*x-147)/(x*log(1/3*x)^2+(-14*x^4+490*x^3-2198*x^2+2940*x)*log(1/3*x)+49 
*x^7-3430*x^6+75411*x^5-559090*x^4+1928101*x^3-3231060*x^2+2160900*x),x, a 
lgorithm=\
 

output

-21*(x^2 - 5*x + 7)/(7*x^3 - 245*x^2 + 1099*x - log(1/3*x) - 1470)
 

3.1.72.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-147+7308 x-10311 x^2+5733 x^3-1470 x^4+147 x^5+\left (-105 x+42 x^2\right ) \log \left (\frac {x}{3}\right )}{2160900 x-3231060 x^2+1928101 x^3-559090 x^4+75411 x^5-3430 x^6+49 x^7+\left (2940 x-2198 x^2+490 x^3-14 x^4\right ) \log \left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right )} \, dx=-\int \frac {\ln \left (\frac {x}{3}\right )\,\left (105\,x-42\,x^2\right )-7308\,x+10311\,x^2-5733\,x^3+1470\,x^4-147\,x^5+147}{2160900\,x+x\,{\ln \left (\frac {x}{3}\right )}^2+\ln \left (\frac {x}{3}\right )\,\left (-14\,x^4+490\,x^3-2198\,x^2+2940\,x\right )-3231060\,x^2+1928101\,x^3-559090\,x^4+75411\,x^5-3430\,x^6+49\,x^7} \,d x \]

input

int(-(log(x/3)*(105*x - 42*x^2) - 7308*x + 10311*x^2 - 5733*x^3 + 1470*x^4 
 - 147*x^5 + 147)/(2160900*x + x*log(x/3)^2 + log(x/3)*(2940*x - 2198*x^2 
+ 490*x^3 - 14*x^4) - 3231060*x^2 + 1928101*x^3 - 559090*x^4 + 75411*x^5 - 
 3430*x^6 + 49*x^7),x)
 

output

-int((log(x/3)*(105*x - 42*x^2) - 7308*x + 10311*x^2 - 5733*x^3 + 1470*x^4 
 - 147*x^5 + 147)/(2160900*x + x*log(x/3)^2 + log(x/3)*(2940*x - 2198*x^2 
+ 490*x^3 - 14*x^4) - 3231060*x^2 + 1928101*x^3 - 559090*x^4 + 75411*x^5 - 
 3430*x^6 + 49*x^7), x)