Integrand size = 179, antiderivative size = 25 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4}{x \left (5+\log (9 x)+\frac {\log (2+x)}{7-x}\right )} \]
Time = 0.86 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=-\frac {4 (-7+x)}{x (35-5 x-(-7+x) \log (9 x)+\log (2+x))} \]
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]
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 |
\(\displaystyle -4 \left (49 \int \frac {1}{x^2 (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx-7 \int \frac {1}{x^2 (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)}dx+6 \int \frac {1}{(\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx-\frac {91}{2} \int \frac {1}{x (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx-\frac {9}{2} \int \frac {1}{(x+2) (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx+\int \frac {\log (9 x)}{(\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx-7 \int \frac {\log (9 x)}{x (\log (9 x) x+5 x-7 \log (9 x)-\log (x+2)-35)^2}dx\right )\) |
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]
3.19.41.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.40 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36
method | result | size |
risch | \(\frac {-28+4 x}{x \left (x \ln \left (9 x \right )+5 x -7 \ln \left (9 x \right )-\ln \left (2+x \right )-35\right )}\) | \(34\) |
parallelrisch | \(\frac {-28+4 x}{x \left (x \ln \left (9 x \right )+5 x -7 \ln \left (9 x \right )-\ln \left (2+x \right )-35\right )}\) | \(35\) |
default | \(\frac {-28+4 x}{x \left (2 x \ln \left (3\right )+x \ln \left (x \right )-14 \ln \left (3\right )+5 x -\ln \left (2+x \right )-7 \ln \left (x \right )-35\right )}\) | \(39\) |
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)
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4 \, {\left (x - 7\right )}}{5 \, x^{2} + {\left (x^{2} - 7 \, x\right )} \log \left (9 \, x\right ) - x \log \left (x + 2\right ) - 35 \, x} \]
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=\
Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {28 - 4 x}{- x^{2} \log {\left (9 x \right )} - 5 x^{2} + 7 x \log {\left (9 x \right )} + x \log {\left (x + 2 \right )} + 35 x} \]
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)
Time = 0.32 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4 \, {\left (x - 7\right )}}{x^{2} {\left (2 \, \log \left (3\right ) + 5\right )} - 7 \, x {\left (2 \, \log \left (3\right ) + 5\right )} - x \log \left (x + 2\right ) + {\left (x^{2} - 7 \, x\right )} \log \left (x\right )} \]
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=\
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).
Time = 0.35 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.36 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=\frac {4 \, {\left (x - 7\right )}}{{\left (x + 2\right )}^{2} \log \left (9 \, x\right ) + 5 \, {\left (x + 2\right )}^{2} - 11 \, {\left (x + 2\right )} \log \left (9 \, x\right ) - {\left (x + 2\right )} \log \left (x + 2\right ) - 55 \, x + 18 \, \log \left (9 \, x\right ) + 2 \, \log \left (x + 2\right ) - 20} \]
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=\
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)
Time = 11.58 (sec) , antiderivative size = 142, normalized size of antiderivative = 5.68 \[ \int \frac {-2352-532 x+292 x^2-24 x^3+\left (-392-84 x+48 x^2-4 x^3\right ) \log (9 x)+(-56-28 x) \log (2+x)}{2450 x^2+525 x^3-300 x^4+25 x^5+\left (98 x^2+21 x^3-12 x^4+x^5\right ) \log ^2(9 x)+\left (140 x^2+50 x^3-10 x^4\right ) \log (2+x)+\left (2 x^2+x^3\right ) \log ^2(2+x)+\log (9 x) \left (980 x^2+210 x^3-120 x^4+10 x^5+\left (28 x^2+10 x^3-2 x^4\right ) \log (2+x)\right )} \, dx=-\frac {4\,{\left (x^2+2\,x\right )}^2\,\left (-x^4+20\,x^3-119\,x^2+98\,x+686\right )+4\,\ln \left (x+2\right )\,{\left (x^2+2\,x\right )}^2\,\left (-x^3+5\,x^2+14\,x\right )}{x^2\,\left (x+2\right )\,\left (5\,x-\ln \left (x+2\right )+\ln \left (9\,x\right )\,\left (x-7\right )-35\right )\,\left (196\,x+4\,x^2\,\ln \left (x+2\right )+4\,x^3\,\ln \left (x+2\right )+x^4\,\ln \left (x+2\right )+154\,x^2+2\,x^3-11\,x^4+x^5\right )} \]
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)