\(\int \frac {-2352-532 x+292 x^2-24 x^3+(-392-84 x+48 x^2-4 x^3) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+(98 x^2+21 x^3-12 x^4+x^5) \log ^2(9 x)+(140 x^2+50 x^3-10 x^4) \log (2+x)+(2 x^2+x^3) \log ^2(2+x)+\log (9 x) (980 x^2+210 x^3-120 x^4+10 x^5+(28 x^2+10 x^3-2 x^4) \log (2+x))} \, dx\) [1841]

Optimal result

Integrand size = 179, antiderivative size = 25 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4}{x \left (5+\log (9 x)+\frac {\log (2+x)}{7-x}\right )} \] Output:

4/x/(5+ln(9*x)+ln(2+x)/(-x+7))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=-\frac {4 (-7+x)}{x (35-5 x-(-7+x) \log (9 x)+\log (2+x))} \] Input:

Integrate[(-2352 - 532*x + 292*x^2 - 24*x^3 + (-392 - 84*x + 48*x^2 - 4*x^ 
3)*Log[9*x] + (-56 - 28*x)*Log[2 + x])/(2450*x^2 + 525*x^3 - 300*x^4 + 25* 
x^5 + (98*x^2 + 21*x^3 - 12*x^4 + x^5)*Log[9*x]^2 + (140*x^2 + 50*x^3 - 10 
*x^4)*Log[2 + x] + (2*x^2 + x^3)*Log[2 + x]^2 + Log[9*x]*(980*x^2 + 210*x^ 
3 - 120*x^4 + 10*x^5 + (28*x^2 + 10*x^3 - 2*x^4)*Log[2 + x])),x]
 

Output:

(-4*(-7 + x))/(x*(35 - 5*x - (-7 + x)*Log[9*x] + Log[2 + x]))
 

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.

\(\displaystyle \int \frac {-24 x^3+292 x^2+\left (-4 x^3+48 x^2-84 x-392\right ) \log (9 x)-532 x+(-28 x-56) \log (x+2)-2352}{25 x^5-300 x^4+525 x^3+2450 x^2+\left (x^3+2 x^2\right ) \log ^2(x+2)+\left (-10 x^4+50 x^3+140 x^2\right ) \log (x+2)+\left (x^5-12 x^4+21 x^3+98 x^2\right ) \log ^2(9 x)+\log (9 x) \left (10 x^5-120 x^4+210 x^3+980 x^2+\left (-2 x^4+10 x^3+28 x^2\right ) \log (x+2)\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {4 \left (-6 x^3+73 x^2-133 x-(x-7)^2 (x+2) \log (9 x)-7 (x+2) \log (x+2)-588\right )}{x^2 (x+2) (5 (x-7)+(x-7) \log (9 x)-\log (x+2))^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \int -\frac {6 x^3-73 x^2+133 x+(7-x)^2 (x+2) \log (9 x)+7 (x+2) \log (x+2)+588}{x^2 (x+2) (\log (9 x) (7-x)+5 (7-x)+\log (x+2))^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -4 \int \frac {6 x^3-73 x^2+133 x+(7-x)^2 (x+2) \log (9 x)+7 (x+2) \log (x+2)+588}{x^2 (x+2) (\log (9 x) (7-x)+5 (7-x)+\log (x+2))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {(x-7) \left (\log (9 x) x^2+6 x^2+2 \log (9 x) x+4 x-14\right )}{x^2 (x+2) (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}-\frac {7}{x^2 (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -4 \left (49 \int \frac {1}{x^2 (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx-7 \int \frac {1}{x^2 (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)}dx+6 \int \frac {1}{(\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx-\frac {91}{2} \int \frac {1}{x (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx-\frac {9}{2} \int \frac {1}{(x+2) (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx+\int \frac {\log (9 x)}{(\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx-7 \int \frac {\log (9 x)}{x (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx\right )\)

Input:

Int[(-2352 - 532*x + 292*x^2 - 24*x^3 + (-392 - 84*x + 48*x^2 - 4*x^3)*Log 
[9*x] + (-56 - 28*x)*Log[2 + x])/(2450*x^2 + 525*x^3 - 300*x^4 + 25*x^5 + 
(98*x^2 + 21*x^3 - 12*x^4 + x^5)*Log[9*x]^2 + (140*x^2 + 50*x^3 - 10*x^4)* 
Log[2 + x] + (2*x^2 + x^3)*Log[2 + x]^2 + Log[9*x]*(980*x^2 + 210*x^3 - 12 
0*x^4 + 10*x^5 + (28*x^2 + 10*x^3 - 2*x^4)*Log[2 + x])),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.67 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36

method	result	size
risch	\(\frac {-28+4 x}{x \left (\ln \left (9 x \right ) x -7 \ln \left (9 x \right )-\ln \left (2+x \right )+5 x -35\right )}\)	\(34\)
parallelrisch	\(\frac {-28+4 x}{x \left (\ln \left (9 x \right ) x -7 \ln \left (9 x \right )-\ln \left (2+x \right )+5 x -35\right )}\)	\(35\)
default	\(\frac {-28+4 x}{x \left (2 x \ln \left (3\right )+x \ln \left (x \right )-14 \ln \left (3\right )+5 x -\ln \left (2+x \right )-7 \ln \left (x \right )-35\right )}\)	\(39\)

Input:

int(((-4*x^3+48*x^2-84*x-392)*ln(9*x)+(-28*x-56)*ln(2+x)-24*x^3+292*x^2-53 
2*x-2352)/((x^5-12*x^4+21*x^3+98*x^2)*ln(9*x)^2+((-2*x^4+10*x^3+28*x^2)*ln 
(2+x)+10*x^5-120*x^4+210*x^3+980*x^2)*ln(9*x)+(x^3+2*x^2)*ln(2+x)^2+(-10*x 
^4+50*x^3+140*x^2)*ln(2+x)+25*x^5-300*x^4+525*x^3+2450*x^2),x,method=_RETU 
RNVERBOSE)
 

Output:

4*(-7+x)/x/(ln(9*x)*x-7*ln(9*x)-ln(2+x)+5*x-35)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4 \, {\left (x - 7\right )}}{5 \, x^{2} + {\left (x^{2} - 7 \, x\right )} \log \left (9 \, x\right ) - x \log \left (x + 2\right ) - 35 \, x} \] Input:

integrate(((-4*x^3+48*x^2-84*x-392)*log(9*x)+(-28*x-56)*log(2+x)-24*x^3+29 
2*x^2-532*x-2352)/((x^5-12*x^4+21*x^3+98*x^2)*log(9*x)^2+((-2*x^4+10*x^3+2 
8*x^2)*log(2+x)+10*x^5-120*x^4+210*x^3+980*x^2)*log(9*x)+(x^3+2*x^2)*log(2 
+x)^2+(-10*x^4+50*x^3+140*x^2)*log(2+x)+25*x^5-300*x^4+525*x^3+2450*x^2),x 
, algorithm="fricas")
 

Output:

4*(x - 7)/(5*x^2 + (x^2 - 7*x)*log(9*x) - x*log(x + 2) - 35*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {28 - 4 x}{- x^{2} \log {\left (9 x \right )} - 5 x^{2} + 7 x \log {\left (9 x \right )} + x \log {\left (x + 2 \right )} + 35 x} \] Input:

integrate(((-4*x**3+48*x**2-84*x-392)*ln(9*x)+(-28*x-56)*ln(2+x)-24*x**3+2 
92*x**2-532*x-2352)/((x**5-12*x**4+21*x**3+98*x**2)*ln(9*x)**2+((-2*x**4+1 
0*x**3+28*x**2)*ln(2+x)+10*x**5-120*x**4+210*x**3+980*x**2)*ln(9*x)+(x**3+ 
2*x**2)*ln(2+x)**2+(-10*x**4+50*x**3+140*x**2)*ln(2+x)+25*x**5-300*x**4+52 
5*x**3+2450*x**2),x)
 

Output:

(28 - 4*x)/(-x**2*log(9*x) - 5*x**2 + 7*x*log(9*x) + x*log(x + 2) + 35*x)
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4 \, {\left (x - 7\right )}}{x^{2} {\left (2 \, \log \left (3\right ) + 5\right )} - 7 \, x {\left (2 \, \log \left (3\right ) + 5\right )} - x \log \left (x + 2\right ) + {\left (x^{2} - 7 \, x\right )} \log \left (x\right )} \] Input:

integrate(((-4*x^3+48*x^2-84*x-392)*log(9*x)+(-28*x-56)*log(2+x)-24*x^3+29 
2*x^2-532*x-2352)/((x^5-12*x^4+21*x^3+98*x^2)*log(9*x)^2+((-2*x^4+10*x^3+2 
8*x^2)*log(2+x)+10*x^5-120*x^4+210*x^3+980*x^2)*log(9*x)+(x^3+2*x^2)*log(2 
+x)^2+(-10*x^4+50*x^3+140*x^2)*log(2+x)+25*x^5-300*x^4+525*x^3+2450*x^2),x 
, algorithm="maxima")
 

Output:

4*(x - 7)/(x^2*(2*log(3) + 5) - 7*x*(2*log(3) + 5) - x*log(x + 2) + (x^2 - 
 7*x)*log(x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).

Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.36 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4 \, {\left (x - 7\right )}}{{\left (x + 2\right )}^{2} \log \left (9 \, x\right ) + 5 \, {\left (x + 2\right )}^{2} - 11 \, {\left (x + 2\right )} \log \left (9 \, x\right ) - {\left (x + 2\right )} \log \left (x + 2\right ) - 55 \, x + 18 \, \log \left (9 \, x\right ) + 2 \, \log \left (x + 2\right ) - 20} \] Input:

integrate(((-4*x^3+48*x^2-84*x-392)*log(9*x)+(-28*x-56)*log(2+x)-24*x^3+29 
2*x^2-532*x-2352)/((x^5-12*x^4+21*x^3+98*x^2)*log(9*x)^2+((-2*x^4+10*x^3+2 
8*x^2)*log(2+x)+10*x^5-120*x^4+210*x^3+980*x^2)*log(9*x)+(x^3+2*x^2)*log(2 
+x)^2+(-10*x^4+50*x^3+140*x^2)*log(2+x)+25*x^5-300*x^4+525*x^3+2450*x^2),x 
, algorithm="giac")
 

Output:

4*(x - 7)/((x + 2)^2*log(9*x) + 5*(x + 2)^2 - 11*(x + 2)*log(9*x) - (x + 2 
)*log(x + 2) - 55*x + 18*log(9*x) + 2*log(x + 2) - 20)
 

Mupad [B] (verification not implemented)

Time = 4.06 (sec) , antiderivative size = 142, normalized size of antiderivative = 5.68 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=-\frac {4\,{\left (x^2+2\,x\right )}^2\,\left (-x^4+20\,x^3-119\,x^2+98\,x+686\right )+4\,\ln \left (x+2\right )\,{\left (x^2+2\,x\right )}^2\,\left (-x^3+5\,x^2+14\,x\right )}{x^2\,\left (x+2\right )\,\left (5\,x-\ln \left (x+2\right )+\ln \left (9\,x\right )\,\left (x-7\right )-35\right )\,\left (196\,x+4\,x^2\,\ln \left (x+2\right )+4\,x^3\,\ln \left (x+2\right )+x^4\,\ln \left (x+2\right )+154\,x^2+2\,x^3-11\,x^4+x^5\right )} \] Input:

int(-(532*x + log(9*x)*(84*x - 48*x^2 + 4*x^3 + 392) - 292*x^2 + 24*x^3 + 
log(x + 2)*(28*x + 56) + 2352)/(log(x + 2)^2*(2*x^2 + x^3) + log(9*x)*(log 
(x + 2)*(28*x^2 + 10*x^3 - 2*x^4) + 980*x^2 + 210*x^3 - 120*x^4 + 10*x^5) 
+ log(x + 2)*(140*x^2 + 50*x^3 - 10*x^4) + 2450*x^2 + 525*x^3 - 300*x^4 + 
25*x^5 + log(9*x)^2*(98*x^2 + 21*x^3 - 12*x^4 + x^5)),x)
 

Output:

-(4*(2*x + x^2)^2*(98*x - 119*x^2 + 20*x^3 - x^4 + 686) + 4*log(x + 2)*(2* 
x + x^2)^2*(14*x + 5*x^2 - x^3))/(x^2*(x + 2)*(5*x - log(x + 2) + log(9*x) 
*(x - 7) - 35)*(196*x + 4*x^2*log(x + 2) + 4*x^3*log(x + 2) + x^4*log(x + 
2) + 154*x^2 + 2*x^3 - 11*x^4 + x^5))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {-4 x +28}{x \left (\mathrm {log}\left (x +2\right )-\mathrm {log}\left (9 x \right ) x +7 \,\mathrm {log}\left (9 x \right )-5 x +35\right )} \] Input:

int(((-4*x^3+48*x^2-84*x-392)*log(9*x)+(-28*x-56)*log(2+x)-24*x^3+292*x^2- 
532*x-2352)/((x^5-12*x^4+21*x^3+98*x^2)*log(9*x)^2+((-2*x^4+10*x^3+28*x^2) 
*log(2+x)+10*x^5-120*x^4+210*x^3+980*x^2)*log(9*x)+(x^3+2*x^2)*log(2+x)^2+ 
(-10*x^4+50*x^3+140*x^2)*log(2+x)+25*x^5-300*x^4+525*x^3+2450*x^2),x)
 

Output:

(4*( - x + 7))/(x*(log(x + 2) - log(9*x)*x + 7*log(9*x) - 5*x + 35))