Integrand size = 129, antiderivative size = 32 \[ \int \frac {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+\left (-8 x^2+120 x^3+108 x^4+24 x^5\right ) \log (x)+12 x^4 \log ^2(x)}{1-20 x+82 x^2+176 x^3+121 x^4+36 x^5+4 x^6+\left (-4 x^2+40 x^3+36 x^4+8 x^5\right ) \log (x)+4 x^4 \log ^2(x)} \, dx=x \left (5-\frac {2 x}{x+\frac {2}{2 x-4 x \left (5+\frac {5}{x}+x+\log (x)\right )}}\right ) \] Output:
(5-2*x/(x+2/(2*x-4*(ln(x)+5+x+5/x)*x)))*x
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+\left (-8 x^2+120 x^3+108 x^4+24 x^5\right ) \log (x)+12 x^4 \log ^2(x)}{1-20 x+82 x^2+176 x^3+121 x^4+36 x^5+4 x^6+\left (-4 x^2+40 x^3+36 x^4+8 x^5\right ) \log (x)+4 x^4 \log ^2(x)} \, dx=3 x-\frac {2 x}{-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)} \] Input:
Integrate[(5 - 60*x + 268*x^2 + 536*x^3 + 363*x^4 + 108*x^5 + 12*x^6 + (-8 *x^2 + 120*x^3 + 108*x^4 + 24*x^5)*Log[x] + 12*x^4*Log[x]^2)/(1 - 20*x + 8 2*x^2 + 176*x^3 + 121*x^4 + 36*x^5 + 4*x^6 + (-4*x^2 + 40*x^3 + 36*x^4 + 8 *x^5)*Log[x] + 4*x^4*Log[x]^2),x]
Output:
3*x - (2*x)/(-1 + 10*x + 9*x^2 + 2*x^3 + 2*x^2*Log[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.
\(\displaystyle \int \frac {12 x^6+108 x^5+363 x^4+12 x^4 \log ^2(x)+536 x^3+268 x^2+\left (24 x^5+108 x^4+120 x^3-8 x^2\right ) \log (x)-60 x+5}{4 x^6+36 x^5+121 x^4+4 x^4 \log ^2(x)+176 x^3+82 x^2+\left (8 x^5+36 x^4+40 x^3-4 x^2\right ) \log (x)-20 x+1} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {12 x^6+108 x^5+363 x^4+12 x^4 \log ^2(x)+536 x^3+268 x^2+\left (24 x^5+108 x^4+120 x^3-8 x^2\right ) \log (x)-60 x+5}{\left (-2 x^3-9 x^2-2 x^2 \log (x)-10 x+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 \left (x^3+x^2-5 x+1\right )}{\left (2 x^3+9 x^2+2 x^2 \log (x)+10 x-1\right )^2}+\frac {2}{2 x^3+9 x^2+2 x^2 \log (x)+10 x-1}+3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {1}{\left (2 x^3+2 \log (x) x^2+9 x^2+10 x-1\right )^2}dx-20 \int \frac {x}{\left (2 x^3+2 \log (x) x^2+9 x^2+10 x-1\right )^2}dx+4 \int \frac {x^2}{\left (2 x^3+2 \log (x) x^2+9 x^2+10 x-1\right )^2}dx+4 \int \frac {x^3}{\left (2 x^3+2 \log (x) x^2+9 x^2+10 x-1\right )^2}dx+2 \int \frac {1}{2 x^3+2 \log (x) x^2+9 x^2+10 x-1}dx+3 x\) |
Input:
Int[(5 - 60*x + 268*x^2 + 536*x^3 + 363*x^4 + 108*x^5 + 12*x^6 + (-8*x^2 + 120*x^3 + 108*x^4 + 24*x^5)*Log[x] + 12*x^4*Log[x]^2)/(1 - 20*x + 82*x^2 + 176*x^3 + 121*x^4 + 36*x^5 + 4*x^6 + (-4*x^2 + 40*x^3 + 36*x^4 + 8*x^5)* Log[x] + 4*x^4*Log[x]^2),x]
Output:
$Aborted
Time = 1.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00
method | result | size |
risch | \(3 x -\frac {2 x}{2 x^{2} \ln \left (x \right )+2 x^{3}+9 x^{2}+10 x -1}\) | \(32\) |
default | \(\frac {-5 x +27 x^{3}+30 x^{2}+6 x^{4}+6 x^{3} \ln \left (x \right )}{2 x^{2} \ln \left (x \right )+2 x^{3}+9 x^{2}+10 x -1}\) | \(52\) |
parallelrisch | \(\frac {12 x^{4}+12 x^{3} \ln \left (x \right )+54 x^{3}+60 x^{2}-10 x}{4 x^{2} \ln \left (x \right )+4 x^{3}+18 x^{2}+20 x -2}\) | \(53\) |
norman | \(\frac {-140 x -\frac {183 x^{2}}{2}-27 x^{2} \ln \left (x \right )+6 x^{4}+6 x^{3} \ln \left (x \right )+\frac {27}{2}}{2 x^{2} \ln \left (x \right )+2 x^{3}+9 x^{2}+10 x -1}\) | \(55\) |
Input:
int((12*x^4*ln(x)^2+(24*x^5+108*x^4+120*x^3-8*x^2)*ln(x)+12*x^6+108*x^5+36 3*x^4+536*x^3+268*x^2-60*x+5)/(4*x^4*ln(x)^2+(8*x^5+36*x^4+40*x^3-4*x^2)*l n(x)+4*x^6+36*x^5+121*x^4+176*x^3+82*x^2-20*x+1),x,method=_RETURNVERBOSE)
Output:
3*x-2*x/(2*x^2*ln(x)+2*x^3+9*x^2+10*x-1)
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+\left (-8 x^2+120 x^3+108 x^4+24 x^5\right ) \log (x)+12 x^4 \log ^2(x)}{1-20 x+82 x^2+176 x^3+121 x^4+36 x^5+4 x^6+\left (-4 x^2+40 x^3+36 x^4+8 x^5\right ) \log (x)+4 x^4 \log ^2(x)} \, dx=\frac {6 \, x^{4} + 6 \, x^{3} \log \left (x\right ) + 27 \, x^{3} + 30 \, x^{2} - 5 \, x}{2 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + 9 \, x^{2} + 10 \, x - 1} \] Input:
integrate((12*x^4*log(x)^2+(24*x^5+108*x^4+120*x^3-8*x^2)*log(x)+12*x^6+10 8*x^5+363*x^4+536*x^3+268*x^2-60*x+5)/(4*x^4*log(x)^2+(8*x^5+36*x^4+40*x^3 -4*x^2)*log(x)+4*x^6+36*x^5+121*x^4+176*x^3+82*x^2-20*x+1),x, algorithm="f ricas")
Output:
(6*x^4 + 6*x^3*log(x) + 27*x^3 + 30*x^2 - 5*x)/(2*x^3 + 2*x^2*log(x) + 9*x ^2 + 10*x - 1)
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+\left (-8 x^2+120 x^3+108 x^4+24 x^5\right ) \log (x)+12 x^4 \log ^2(x)}{1-20 x+82 x^2+176 x^3+121 x^4+36 x^5+4 x^6+\left (-4 x^2+40 x^3+36 x^4+8 x^5\right ) \log (x)+4 x^4 \log ^2(x)} \, dx=3 x - \frac {2 x}{2 x^{3} + 2 x^{2} \log {\left (x \right )} + 9 x^{2} + 10 x - 1} \] Input:
integrate((12*x**4*ln(x)**2+(24*x**5+108*x**4+120*x**3-8*x**2)*ln(x)+12*x* *6+108*x**5+363*x**4+536*x**3+268*x**2-60*x+5)/(4*x**4*ln(x)**2+(8*x**5+36 *x**4+40*x**3-4*x**2)*ln(x)+4*x**6+36*x**5+121*x**4+176*x**3+82*x**2-20*x+ 1),x)
Output:
3*x - 2*x/(2*x**3 + 2*x**2*log(x) + 9*x**2 + 10*x - 1)
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+\left (-8 x^2+120 x^3+108 x^4+24 x^5\right ) \log (x)+12 x^4 \log ^2(x)}{1-20 x+82 x^2+176 x^3+121 x^4+36 x^5+4 x^6+\left (-4 x^2+40 x^3+36 x^4+8 x^5\right ) \log (x)+4 x^4 \log ^2(x)} \, dx=\frac {6 \, x^{4} + 6 \, x^{3} \log \left (x\right ) + 27 \, x^{3} + 30 \, x^{2} - 5 \, x}{2 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + 9 \, x^{2} + 10 \, x - 1} \] Input:
integrate((12*x^4*log(x)^2+(24*x^5+108*x^4+120*x^3-8*x^2)*log(x)+12*x^6+10 8*x^5+363*x^4+536*x^3+268*x^2-60*x+5)/(4*x^4*log(x)^2+(8*x^5+36*x^4+40*x^3 -4*x^2)*log(x)+4*x^6+36*x^5+121*x^4+176*x^3+82*x^2-20*x+1),x, algorithm="m axima")
Output:
(6*x^4 + 6*x^3*log(x) + 27*x^3 + 30*x^2 - 5*x)/(2*x^3 + 2*x^2*log(x) + 9*x ^2 + 10*x - 1)
Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+\left (-8 x^2+120 x^3+108 x^4+24 x^5\right ) \log (x)+12 x^4 \log ^2(x)}{1-20 x+82 x^2+176 x^3+121 x^4+36 x^5+4 x^6+\left (-4 x^2+40 x^3+36 x^4+8 x^5\right ) \log (x)+4 x^4 \log ^2(x)} \, dx=3 \, x - \frac {2 \, x}{2 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + 9 \, x^{2} + 10 \, x - 1} \] Input:
integrate((12*x^4*log(x)^2+(24*x^5+108*x^4+120*x^3-8*x^2)*log(x)+12*x^6+10 8*x^5+363*x^4+536*x^3+268*x^2-60*x+5)/(4*x^4*log(x)^2+(8*x^5+36*x^4+40*x^3 -4*x^2)*log(x)+4*x^6+36*x^5+121*x^4+176*x^3+82*x^2-20*x+1),x, algorithm="g iac")
Output:
3*x - 2*x/(2*x^3 + 2*x^2*log(x) + 9*x^2 + 10*x - 1)
Time = 3.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+\left (-8 x^2+120 x^3+108 x^4+24 x^5\right ) \log (x)+12 x^4 \log ^2(x)}{1-20 x+82 x^2+176 x^3+121 x^4+36 x^5+4 x^6+\left (-4 x^2+40 x^3+36 x^4+8 x^5\right ) \log (x)+4 x^4 \log ^2(x)} \, dx=3\,x-\frac {2\,x}{10\,x+2\,x^2\,\ln \left (x\right )+9\,x^2+2\,x^3-1} \] Input:
int((12*x^4*log(x)^2 - 60*x + log(x)*(120*x^3 - 8*x^2 + 108*x^4 + 24*x^5) + 268*x^2 + 536*x^3 + 363*x^4 + 108*x^5 + 12*x^6 + 5)/(4*x^4*log(x)^2 - 20 *x + log(x)*(40*x^3 - 4*x^2 + 36*x^4 + 8*x^5) + 82*x^2 + 176*x^3 + 121*x^4 + 36*x^5 + 4*x^6 + 1),x)
Output:
3*x - (2*x)/(10*x + 2*x^2*log(x) + 9*x^2 + 2*x^3 - 1)
Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \frac {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+\left (-8 x^2+120 x^3+108 x^4+24 x^5\right ) \log (x)+12 x^4 \log ^2(x)}{1-20 x+82 x^2+176 x^3+121 x^4+36 x^5+4 x^6+\left (-4 x^2+40 x^3+36 x^4+8 x^5\right ) \log (x)+4 x^4 \log ^2(x)} \, dx=\frac {18 \,\mathrm {log}\left (x \right ) x^{3}-20 \,\mathrm {log}\left (x \right ) x^{2}+18 x^{4}+61 x^{3}-115 x +10}{6 \,\mathrm {log}\left (x \right ) x^{2}+6 x^{3}+27 x^{2}+30 x -3} \] Input:
int((12*x^4*log(x)^2+(24*x^5+108*x^4+120*x^3-8*x^2)*log(x)+12*x^6+108*x^5+ 363*x^4+536*x^3+268*x^2-60*x+5)/(4*x^4*log(x)^2+(8*x^5+36*x^4+40*x^3-4*x^2 )*log(x)+4*x^6+36*x^5+121*x^4+176*x^3+82*x^2-20*x+1),x)
Output:
(18*log(x)*x**3 - 20*log(x)*x**2 + 18*x**4 + 61*x**3 - 115*x + 10)/(3*(2*l og(x)*x**2 + 2*x**3 + 9*x**2 + 10*x - 1))