Integrand size = 125, antiderivative size = 28 \[ \int \frac {-6480-4320 x-924 x^2-70 x^3-x^4+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log (x)}{20736-24480 x-3143 x^2+5256 x^3+1806 x^4+216 x^5+9 x^6+\left (-10368+2664 x+4344 x^2+1250 x^3+144 x^4+6 x^5\right ) \log (x)+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x}{-4-x+\frac {\left (2 x+\frac {x}{6+x}\right )^2}{x}+\log (x)} \] Output:
x/(ln(x)-4+(2*x+x/(6+x))^2/x-x)
Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-6480-4320 x-924 x^2-70 x^3-x^4+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log (x)}{20736-24480 x-3143 x^2+5256 x^3+1806 x^4+216 x^5+9 x^6+\left (-10368+2664 x+4344 x^2+1250 x^3+144 x^4+6 x^5\right ) \log (x)+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x (6+x)^2}{-144+85 x+36 x^2+3 x^3+(6+x)^2 \log (x)} \] Input:
Integrate[(-6480 - 4320*x - 924*x^2 - 70*x^3 - x^4 + (1296 + 864*x + 216*x ^2 + 24*x^3 + x^4)*Log[x])/(20736 - 24480*x - 3143*x^2 + 5256*x^3 + 1806*x ^4 + 216*x^5 + 9*x^6 + (-10368 + 2664*x + 4344*x^2 + 1250*x^3 + 144*x^4 + 6*x^5)*Log[x] + (1296 + 864*x + 216*x^2 + 24*x^3 + x^4)*Log[x]^2),x]
Output:
(x*(6 + x)^2)/(-144 + 85*x + 36*x^2 + 3*x^3 + (6 + 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.
| \(\displaystyle \int \frac {-x^4-70 x^3-924 x^2+\left (x^4+24 x^3+216 x^2+864 x+1296\right ) \log (x)-4320 x-6480}{9 x^6+216 x^5+1806 x^4+5256 x^3-3143 x^2+\left (x^4+24 x^3+216 x^2+864 x+1296\right ) \log ^2(x)+\left (6 x^5+144 x^4+1250 x^3+4344 x^2+2664 x-10368\right ) \log (x)-24480 x+20736} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+6) \left (-x^3-64 x^2-540 x+(x+6)^3 \log (x)-1080\right )}{\left (-3 x^3-36 x^2-85 x-(x+6)^2 \log (x)+144\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x+6)^2}{3 x^3+36 x^2+x^2 \log (x)+85 x+12 x \log (x)+36 \log (x)-144}+\frac {-3 x^5-73 x^4-695 x^3-3096 x^2-5652 x-1296}{\left (3 x^3+36 x^2+x^2 \log (x)+85 x+12 x \log (x)+36 \log (x)-144\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -1296 \int \frac {1}{\left (3 x^3+\log (x) x^2+36 x^2+12 \log (x) x+85 x+36 \log (x)-144\right )^2}dx-5652 \int \frac {x}{\left (3 x^3+\log (x) x^2+36 x^2+12 \log (x) x+85 x+36 \log (x)-144\right )^2}dx-3096 \int \frac {x^2}{\left (3 x^3+\log (x) x^2+36 x^2+12 \log (x) x+85 x+36 \log (x)-144\right )^2}dx-695 \int \frac {x^3}{\left (3 x^3+\log (x) x^2+36 x^2+12 \log (x) x+85 x+36 \log (x)-144\right )^2}dx+36 \int \frac {1}{3 x^3+\log (x) x^2+36 x^2+12 \log (x) x+85 x+36 \log (x)-144}dx+12 \int \frac {x}{3 x^3+\log (x) x^2+36 x^2+12 \log (x) x+85 x+36 \log (x)-144}dx+\int \frac {x^2}{3 x^3+\log (x) x^2+36 x^2+12 \log (x) x+85 x+36 \log (x)-144}dx-3 \int \frac {x^5}{\left (3 x^3+\log (x) x^2+36 x^2+12 \log (x) x+85 x+36 \log (x)-144\right )^2}dx-73 \int \frac {x^4}{\left (3 x^3+\log (x) x^2+36 x^2+12 \log (x) x+85 x+36 \log (x)-144\right )^2}dx\) |
Input:
Int[(-6480 - 4320*x - 924*x^2 - 70*x^3 - x^4 + (1296 + 864*x + 216*x^2 + 2 4*x^3 + x^4)*Log[x])/(20736 - 24480*x - 3143*x^2 + 5256*x^3 + 1806*x^4 + 2 16*x^5 + 9*x^6 + (-10368 + 2664*x + 4344*x^2 + 1250*x^3 + 144*x^4 + 6*x^5) *Log[x] + (1296 + 864*x + 216*x^2 + 24*x^3 + x^4)*Log[x]^2),x]
Output:
$Aborted
Time = 0.40 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43
| method | result | size |
| risch | \(\frac {\left (6+x \right )^{2} x}{x^{2} \ln \left (x \right )+3 x^{3}+12 x \ln \left (x \right )+36 x^{2}+36 \ln \left (x \right )+85 x -144}\) | \(40\) |
| default | \(\frac {x^{3}+12 x^{2}+36 x}{x^{2} \ln \left (x \right )+3 x^{3}+12 x \ln \left (x \right )+36 x^{2}+36 \ln \left (x \right )+85 x -144}\) | \(46\) |
| parallelrisch | \(\frac {x^{3}+12 x^{2}+36 x}{x^{2} \ln \left (x \right )+3 x^{3}+12 x \ln \left (x \right )+36 x^{2}+36 \ln \left (x \right )+85 x -144}\) | \(46\) |
| norman | \(\frac {-12 \ln \left (x \right )+\frac {23 x}{3}-4 x \ln \left (x \right )-\frac {x^{2} \ln \left (x \right )}{3}+48}{x^{2} \ln \left (x \right )+3 x^{3}+12 x \ln \left (x \right )+36 x^{2}+36 \ln \left (x \right )+85 x -144}\) | \(55\) |
Input:
int(((x^4+24*x^3+216*x^2+864*x+1296)*ln(x)-x^4-70*x^3-924*x^2-4320*x-6480) /((x^4+24*x^3+216*x^2+864*x+1296)*ln(x)^2+(6*x^5+144*x^4+1250*x^3+4344*x^2 +2664*x-10368)*ln(x)+9*x^6+216*x^5+1806*x^4+5256*x^3-3143*x^2-24480*x+2073 6),x,method=_RETURNVERBOSE)
Output:
(6+x)^2*x/(x^2*ln(x)+3*x^3+12*x*ln(x)+36*x^2+36*ln(x)+85*x-144)
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {-6480-4320 x-924 x^2-70 x^3-x^4+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log (x)}{20736-24480 x-3143 x^2+5256 x^3+1806 x^4+216 x^5+9 x^6+\left (-10368+2664 x+4344 x^2+1250 x^3+144 x^4+6 x^5\right ) \log (x)+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x^{3} + 12 \, x^{2} + 36 \, x}{3 \, x^{3} + 36 \, x^{2} + {\left (x^{2} + 12 \, x + 36\right )} \log \left (x\right ) + 85 \, x - 144} \] Input:
integrate(((x^4+24*x^3+216*x^2+864*x+1296)*log(x)-x^4-70*x^3-924*x^2-4320* x-6480)/((x^4+24*x^3+216*x^2+864*x+1296)*log(x)^2+(6*x^5+144*x^4+1250*x^3+ 4344*x^2+2664*x-10368)*log(x)+9*x^6+216*x^5+1806*x^4+5256*x^3-3143*x^2-244 80*x+20736),x, algorithm="fricas")
Output:
(x^3 + 12*x^2 + 36*x)/(3*x^3 + 36*x^2 + (x^2 + 12*x + 36)*log(x) + 85*x - 144)
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {-6480-4320 x-924 x^2-70 x^3-x^4+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log (x)}{20736-24480 x-3143 x^2+5256 x^3+1806 x^4+216 x^5+9 x^6+\left (-10368+2664 x+4344 x^2+1250 x^3+144 x^4+6 x^5\right ) \log (x)+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x^{3} + 12 x^{2} + 36 x}{3 x^{3} + 36 x^{2} + 85 x + \left (x^{2} + 12 x + 36\right ) \log {\left (x \right )} - 144} \] Input:
integrate(((x**4+24*x**3+216*x**2+864*x+1296)*ln(x)-x**4-70*x**3-924*x**2- 4320*x-6480)/((x**4+24*x**3+216*x**2+864*x+1296)*ln(x)**2+(6*x**5+144*x**4 +1250*x**3+4344*x**2+2664*x-10368)*ln(x)+9*x**6+216*x**5+1806*x**4+5256*x* *3-3143*x**2-24480*x+20736),x)
Output:
(x**3 + 12*x**2 + 36*x)/(3*x**3 + 36*x**2 + 85*x + (x**2 + 12*x + 36)*log( x) - 144)
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {-6480-4320 x-924 x^2-70 x^3-x^4+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log (x)}{20736-24480 x-3143 x^2+5256 x^3+1806 x^4+216 x^5+9 x^6+\left (-10368+2664 x+4344 x^2+1250 x^3+144 x^4+6 x^5\right ) \log (x)+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x^{3} + 12 \, x^{2} + 36 \, x}{3 \, x^{3} + 36 \, x^{2} + {\left (x^{2} + 12 \, x + 36\right )} \log \left (x\right ) + 85 \, x - 144} \] Input:
integrate(((x^4+24*x^3+216*x^2+864*x+1296)*log(x)-x^4-70*x^3-924*x^2-4320* x-6480)/((x^4+24*x^3+216*x^2+864*x+1296)*log(x)^2+(6*x^5+144*x^4+1250*x^3+ 4344*x^2+2664*x-10368)*log(x)+9*x^6+216*x^5+1806*x^4+5256*x^3-3143*x^2-244 80*x+20736),x, algorithm="maxima")
Output:
(x^3 + 12*x^2 + 36*x)/(3*x^3 + 36*x^2 + (x^2 + 12*x + 36)*log(x) + 85*x - 144)
Time = 0.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-6480-4320 x-924 x^2-70 x^3-x^4+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log (x)}{20736-24480 x-3143 x^2+5256 x^3+1806 x^4+216 x^5+9 x^6+\left (-10368+2664 x+4344 x^2+1250 x^3+144 x^4+6 x^5\right ) \log (x)+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x^{3} + 12 \, x^{2} + 36 \, x}{3 \, x^{3} + x^{2} \log \left (x\right ) + 36 \, x^{2} + 12 \, x \log \left (x\right ) + 85 \, x + 36 \, \log \left (x\right ) - 144} \] Input:
integrate(((x^4+24*x^3+216*x^2+864*x+1296)*log(x)-x^4-70*x^3-924*x^2-4320* x-6480)/((x^4+24*x^3+216*x^2+864*x+1296)*log(x)^2+(6*x^5+144*x^4+1250*x^3+ 4344*x^2+2664*x-10368)*log(x)+9*x^6+216*x^5+1806*x^4+5256*x^3-3143*x^2-244 80*x+20736),x, algorithm="giac")
Output:
(x^3 + 12*x^2 + 36*x)/(3*x^3 + x^2*log(x) + 36*x^2 + 12*x*log(x) + 85*x + 36*log(x) - 144)
Timed out. \[ \int \frac {-6480-4320 x-924 x^2-70 x^3-x^4+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log (x)}{20736-24480 x-3143 x^2+5256 x^3+1806 x^4+216 x^5+9 x^6+\left (-10368+2664 x+4344 x^2+1250 x^3+144 x^4+6 x^5\right ) \log (x)+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log ^2(x)} \, dx=\int -\frac {4320\,x-\ln \left (x\right )\,\left (x^4+24\,x^3+216\,x^2+864\,x+1296\right )+924\,x^2+70\,x^3+x^4+6480}{{\ln \left (x\right )}^2\,\left (x^4+24\,x^3+216\,x^2+864\,x+1296\right )-24480\,x+\ln \left (x\right )\,\left (6\,x^5+144\,x^4+1250\,x^3+4344\,x^2+2664\,x-10368\right )-3143\,x^2+5256\,x^3+1806\,x^4+216\,x^5+9\,x^6+20736} \,d x \] Input:
int(-(4320*x - log(x)*(864*x + 216*x^2 + 24*x^3 + x^4 + 1296) + 924*x^2 + 70*x^3 + x^4 + 6480)/(log(x)^2*(864*x + 216*x^2 + 24*x^3 + x^4 + 1296) - 2 4480*x + log(x)*(2664*x + 4344*x^2 + 1250*x^3 + 144*x^4 + 6*x^5 - 10368) - 3143*x^2 + 5256*x^3 + 1806*x^4 + 216*x^5 + 9*x^6 + 20736),x)
Output:
int(-(4320*x - log(x)*(864*x + 216*x^2 + 24*x^3 + x^4 + 1296) + 924*x^2 + 70*x^3 + x^4 + 6480)/(log(x)^2*(864*x + 216*x^2 + 24*x^3 + x^4 + 1296) - 2 4480*x + log(x)*(2664*x + 4344*x^2 + 1250*x^3 + 144*x^4 + 6*x^5 - 10368) - 3143*x^2 + 5256*x^3 + 1806*x^4 + 216*x^5 + 9*x^6 + 20736), x)
Time = 0.16 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int \frac {-6480-4320 x-924 x^2-70 x^3-x^4+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log (x)}{20736-24480 x-3143 x^2+5256 x^3+1806 x^4+216 x^5+9 x^6+\left (-10368+2664 x+4344 x^2+1250 x^3+144 x^4+6 x^5\right ) \log (x)+\left (1296+864 x+216 x^2+24 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {-36 \,\mathrm {log}\left (x \right ) x^{2}-432 \,\mathrm {log}\left (x \right ) x -1296 \,\mathrm {log}\left (x \right )-23 x^{3}-276 x^{2}+5184}{85 \,\mathrm {log}\left (x \right ) x^{2}+1020 \,\mathrm {log}\left (x \right ) x +3060 \,\mathrm {log}\left (x \right )+255 x^{3}+3060 x^{2}+7225 x -12240} \] Input:
int(((x^4+24*x^3+216*x^2+864*x+1296)*log(x)-x^4-70*x^3-924*x^2-4320*x-6480 )/((x^4+24*x^3+216*x^2+864*x+1296)*log(x)^2+(6*x^5+144*x^4+1250*x^3+4344*x ^2+2664*x-10368)*log(x)+9*x^6+216*x^5+1806*x^4+5256*x^3-3143*x^2-24480*x+2 0736),x)
Output:
( - 36*log(x)*x**2 - 432*log(x)*x - 1296*log(x) - 23*x**3 - 276*x**2 + 518 4)/(85*(log(x)*x**2 + 12*log(x)*x + 36*log(x) + 3*x**3 + 36*x**2 + 85*x - 144))