Integrand size = 116, antiderivative size = 30 \[ \int \frac {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6+\left (-12220-4798 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)}{1493284 x^2-513240 x^3+66096 x^4-3780 x^5+81 x^6+\left (-4888 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)} \, dx=x+\frac {5}{x \left (-1+\frac {2 \log (x)}{-3+(5+3 (10-x))^2}\right )} \] Output:
x+5/x/(2/((35-3*x)^2-3)*ln(x)-1)
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6+\left (-12220-4798 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)}{1493284 x^2-513240 x^3+66096 x^4-3780 x^5+81 x^6+\left (-4888 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)} \, dx=x+\frac {5 \left (1222-210 x+9 x^2\right )}{x \left (-1222+210 x-9 x^2+2 \log (x)\right )} \] Input:
Integrate[(7454200 - 2564100*x + 1823674*x^2 - 532140*x^3 + 66501*x^4 - 37 80*x^5 + 81*x^6 + (-12220 - 4798*x^2 + 840*x^3 - 36*x^4)*Log[x] + 4*x^2*Lo g[x]^2)/(1493284*x^2 - 513240*x^3 + 66096*x^4 - 3780*x^5 + 81*x^6 + (-4888 *x^2 + 840*x^3 - 36*x^4)*Log[x] + 4*x^2*Log[x]^2),x]
Output:
x + (5*(1222 - 210*x + 9*x^2))/(x*(-1222 + 210*x - 9*x^2 + 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.
\(\displaystyle \int \frac {81 x^6-3780 x^5+66501 x^4-532140 x^3+1823674 x^2+4 x^2 \log ^2(x)+\left (-36 x^4+840 x^3-4798 x^2-12220\right ) \log (x)-2564100 x+7454200}{81 x^6-3780 x^5+66096 x^4-513240 x^3+1493284 x^2+4 x^2 \log ^2(x)+\left (-36 x^4+840 x^3-4888 x^2\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {81 x^6-3780 x^5+66501 x^4-532140 x^3+1823674 x^2+4 x^2 \log ^2(x)+\left (-36 x^4+840 x^3-4798 x^2-12220\right ) \log (x)-2564100 x+7454200}{x^2 \left (9 x^2-210 x-2 \log (x)+1222\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {5 \left (9 x^2-1222\right )}{x^2 \left (9 x^2-210 x-2 \log (x)+1222\right )}+\frac {10 \left (81 x^4-2835 x^3+33039 x^2-128100 x-1222\right )}{x^2 \left (9 x^2-210 x-2 \log (x)+1222\right )^2}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 330390 \int \frac {1}{\left (9 x^2-210 x-2 \log (x)+1222\right )^2}dx-12220 \int \frac {1}{x^2 \left (9 x^2-210 x-2 \log (x)+1222\right )^2}dx-1281000 \int \frac {1}{x \left (9 x^2-210 x-2 \log (x)+1222\right )^2}dx-28350 \int \frac {x}{\left (9 x^2-210 x-2 \log (x)+1222\right )^2}dx+810 \int \frac {x^2}{\left (9 x^2-210 x-2 \log (x)+1222\right )^2}dx-45 \int \frac {1}{9 x^2-210 x-2 \log (x)+1222}dx+6110 \int \frac {1}{x^2 \left (9 x^2-210 x-2 \log (x)+1222\right )}dx+x\) |
Input:
Int[(7454200 - 2564100*x + 1823674*x^2 - 532140*x^3 + 66501*x^4 - 3780*x^5 + 81*x^6 + (-12220 - 4798*x^2 + 840*x^3 - 36*x^4)*Log[x] + 4*x^2*Log[x]^2 )/(1493284*x^2 - 513240*x^3 + 66096*x^4 - 3780*x^5 + 81*x^6 + (-4888*x^2 + 840*x^3 - 36*x^4)*Log[x] + 4*x^2*Log[x]^2),x]
Output:
$Aborted
Time = 1.44 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13
method | result | size |
risch | \(x -\frac {5 \left (9 x^{2}-210 x +1222\right )}{x \left (9 x^{2}-210 x -2 \ln \left (x \right )+1222\right )}\) | \(34\) |
default | \(\frac {6110-1177 x^{2}-1050 x +210 x^{3}-9 x^{4}+2 x^{2} \ln \left (x \right )}{x \left (-9 x^{2}+2 \ln \left (x \right )+210 x -1222\right )}\) | \(48\) |
norman | \(\frac {-6110-3723 x^{2}+\frac {88690 x}{3}-2 x^{2} \ln \left (x \right )-\frac {140 x \ln \left (x \right )}{3}+9 x^{4}}{x \left (9 x^{2}-210 x -2 \ln \left (x \right )+1222\right )}\) | \(48\) |
parallelrisch | \(\frac {18 x^{4}-12220-420 x^{3}-4 x^{2} \ln \left (x \right )+2354 x^{2}+2100 x}{2 x \left (9 x^{2}-210 x -2 \ln \left (x \right )+1222\right )}\) | \(49\) |
Input:
int((4*x^2*ln(x)^2+(-36*x^4+840*x^3-4798*x^2-12220)*ln(x)+81*x^6-3780*x^5+ 66501*x^4-532140*x^3+1823674*x^2-2564100*x+7454200)/(4*x^2*ln(x)^2+(-36*x^ 4+840*x^3-4888*x^2)*ln(x)+81*x^6-3780*x^5+66096*x^4-513240*x^3+1493284*x^2 ),x,method=_RETURNVERBOSE)
Output:
x-5*(9*x^2-210*x+1222)/x/(9*x^2-210*x-2*ln(x)+1222)
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6+\left (-12220-4798 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)}{1493284 x^2-513240 x^3+66096 x^4-3780 x^5+81 x^6+\left (-4888 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)} \, dx=\frac {9 \, x^{4} - 210 \, x^{3} - 2 \, x^{2} \log \left (x\right ) + 1177 \, x^{2} + 1050 \, x - 6110}{9 \, x^{3} - 210 \, x^{2} - 2 \, x \log \left (x\right ) + 1222 \, x} \] Input:
integrate((4*x^2*log(x)^2+(-36*x^4+840*x^3-4798*x^2-12220)*log(x)+81*x^6-3 780*x^5+66501*x^4-532140*x^3+1823674*x^2-2564100*x+7454200)/(4*x^2*log(x)^ 2+(-36*x^4+840*x^3-4888*x^2)*log(x)+81*x^6-3780*x^5+66096*x^4-513240*x^3+1 493284*x^2),x, algorithm="fricas")
Output:
(9*x^4 - 210*x^3 - 2*x^2*log(x) + 1177*x^2 + 1050*x - 6110)/(9*x^3 - 210*x ^2 - 2*x*log(x) + 1222*x)
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6+\left (-12220-4798 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)}{1493284 x^2-513240 x^3+66096 x^4-3780 x^5+81 x^6+\left (-4888 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)} \, dx=x + \frac {45 x^{2} - 1050 x + 6110}{- 9 x^{3} + 210 x^{2} + 2 x \log {\left (x \right )} - 1222 x} \] Input:
integrate((4*x**2*ln(x)**2+(-36*x**4+840*x**3-4798*x**2-12220)*ln(x)+81*x* *6-3780*x**5+66501*x**4-532140*x**3+1823674*x**2-2564100*x+7454200)/(4*x** 2*ln(x)**2+(-36*x**4+840*x**3-4888*x**2)*ln(x)+81*x**6-3780*x**5+66096*x** 4-513240*x**3+1493284*x**2),x)
Output:
x + (45*x**2 - 1050*x + 6110)/(-9*x**3 + 210*x**2 + 2*x*log(x) - 1222*x)
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6+\left (-12220-4798 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)}{1493284 x^2-513240 x^3+66096 x^4-3780 x^5+81 x^6+\left (-4888 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)} \, dx=\frac {9 \, x^{4} - 210 \, x^{3} - 2 \, x^{2} \log \left (x\right ) + 1177 \, x^{2} + 1050 \, x - 6110}{9 \, x^{3} - 210 \, x^{2} - 2 \, x \log \left (x\right ) + 1222 \, x} \] Input:
integrate((4*x^2*log(x)^2+(-36*x^4+840*x^3-4798*x^2-12220)*log(x)+81*x^6-3 780*x^5+66501*x^4-532140*x^3+1823674*x^2-2564100*x+7454200)/(4*x^2*log(x)^ 2+(-36*x^4+840*x^3-4888*x^2)*log(x)+81*x^6-3780*x^5+66096*x^4-513240*x^3+1 493284*x^2),x, algorithm="maxima")
Output:
(9*x^4 - 210*x^3 - 2*x^2*log(x) + 1177*x^2 + 1050*x - 6110)/(9*x^3 - 210*x ^2 - 2*x*log(x) + 1222*x)
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6+\left (-12220-4798 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)}{1493284 x^2-513240 x^3+66096 x^4-3780 x^5+81 x^6+\left (-4888 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)} \, dx=x - \frac {5 \, {\left (9 \, x^{2} - 210 \, x + 1222\right )}}{9 \, x^{3} - 210 \, x^{2} - 2 \, x \log \left (x\right ) + 1222 \, x} \] Input:
integrate((4*x^2*log(x)^2+(-36*x^4+840*x^3-4798*x^2-12220)*log(x)+81*x^6-3 780*x^5+66501*x^4-532140*x^3+1823674*x^2-2564100*x+7454200)/(4*x^2*log(x)^ 2+(-36*x^4+840*x^3-4888*x^2)*log(x)+81*x^6-3780*x^5+66096*x^4-513240*x^3+1 493284*x^2),x, algorithm="giac")
Output:
x - 5*(9*x^2 - 210*x + 1222)/(9*x^3 - 210*x^2 - 2*x*log(x) + 1222*x)
Time = 1.97 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6+\left (-12220-4798 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)}{1493284 x^2-513240 x^3+66096 x^4-3780 x^5+81 x^6+\left (-4888 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)} \, dx=x+\frac {45\,x^2-1050\,x+6110}{x\,\left (210\,x+2\,\ln \left (x\right )-9\,x^2-1222\right )} \] Input:
int((4*x^2*log(x)^2 - log(x)*(4798*x^2 - 840*x^3 + 36*x^4 + 12220) - 25641 00*x + 1823674*x^2 - 532140*x^3 + 66501*x^4 - 3780*x^5 + 81*x^6 + 7454200) /(4*x^2*log(x)^2 - log(x)*(4888*x^2 - 840*x^3 + 36*x^4) + 1493284*x^2 - 51 3240*x^3 + 66096*x^4 - 3780*x^5 + 81*x^6),x)
Output:
x + (45*x^2 - 1050*x + 6110)/(x*(210*x + 2*log(x) - 9*x^2 - 1222))
Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {7454200-2564100 x+1823674 x^2-532140 x^3+66501 x^4-3780 x^5+81 x^6+\left (-12220-4798 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)}{1493284 x^2-513240 x^3+66096 x^4-3780 x^5+81 x^6+\left (-4888 x^2+840 x^3-36 x^4\right ) \log (x)+4 x^2 \log ^2(x)} \, dx=\frac {420 \,\mathrm {log}\left (x \right ) x^{2}+2354 \,\mathrm {log}\left (x \right ) x -1890 x^{4}+33507 x^{3}-1658794 x +1283100}{210 x \left (2 \,\mathrm {log}\left (x \right )-9 x^{2}+210 x -1222\right )} \] Input:
int((4*x^2*log(x)^2+(-36*x^4+840*x^3-4798*x^2-12220)*log(x)+81*x^6-3780*x^ 5+66501*x^4-532140*x^3+1823674*x^2-2564100*x+7454200)/(4*x^2*log(x)^2+(-36 *x^4+840*x^3-4888*x^2)*log(x)+81*x^6-3780*x^5+66096*x^4-513240*x^3+1493284 *x^2),x)
Output:
(420*log(x)*x**2 + 2354*log(x)*x - 1890*x**4 + 33507*x**3 - 1658794*x + 12 83100)/(210*x*(2*log(x) - 9*x**2 + 210*x - 1222))