Integrand size = 82, antiderivative size = 23 \[ \int \frac {-60+16 x-601 x^2+140 x^3-10 x^4-4 x \log (x)}{5000 x^5-1000 x^6+50 x^7+\left (2000 x^3-400 x^4+20 x^5\right ) \log (x)+\left (200 x-40 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {-3+\frac {x}{2}}{(-10+x) \left (5 x^2+\log (x)\right )} \] Output:
(1/2*x-3)/(x-10)/(5*x^2+ln(x))
Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-60+16 x-601 x^2+140 x^3-10 x^4-4 x \log (x)}{5000 x^5-1000 x^6+50 x^7+\left (2000 x^3-400 x^4+20 x^5\right ) \log (x)+\left (200 x-40 x^2+2 x^3\right ) \log ^2(x)} \, dx=-\frac {6-x}{2 (-10+x) \left (5 x^2+\log (x)\right )} \] Input:
Integrate[(-60 + 16*x - 601*x^2 + 140*x^3 - 10*x^4 - 4*x*Log[x])/(5000*x^5 - 1000*x^6 + 50*x^7 + (2000*x^3 - 400*x^4 + 20*x^5)*Log[x] + (200*x - 40* x^2 + 2*x^3)*Log[x]^2),x]
Output:
-1/2*(6 - x)/((-10 + x)*(5*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.
\(\displaystyle \int \frac {-10 x^4+140 x^3-601 x^2+16 x-4 x \log (x)-60}{50 x^7-1000 x^6+5000 x^5+\left (2 x^3-40 x^2+200 x\right ) \log ^2(x)+\left (20 x^5-400 x^4+2000 x^3\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {-10 x^4+140 x^3-601 x^2+16 x-4 x \log (x)-60}{2 (10-x)^2 x \left (5 x^2+\log (x)\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int -\frac {10 x^4-140 x^3+601 x^2+4 \log (x) x-16 x+60}{(10-x)^2 x \left (5 x^2+\log (x)\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {10 x^4-140 x^3+601 x^2+4 \log (x) x-16 x+60}{(10-x)^2 x \left (5 x^2+\log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{2} \int \left (\frac {10 x^3-60 x^2+x-6}{(x-10) x \left (5 x^2+\log (x)\right )^2}+\frac {4}{(x-10)^2 \left (5 x^2+\log (x)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-40 \int \frac {1}{\left (5 x^2+\log (x)\right )^2}dx-\frac {2002}{5} \int \frac {1}{(x-10) \left (5 x^2+\log (x)\right )^2}dx-\frac {3}{5} \int \frac {1}{x \left (5 x^2+\log (x)\right )^2}dx-10 \int \frac {x}{\left (5 x^2+\log (x)\right )^2}dx-4 \int \frac {1}{(x-10)^2 \left (5 x^2+\log (x)\right )}dx\right )\) |
Input:
Int[(-60 + 16*x - 601*x^2 + 140*x^3 - 10*x^4 - 4*x*Log[x])/(5000*x^5 - 100 0*x^6 + 50*x^7 + (2000*x^3 - 400*x^4 + 20*x^5)*Log[x] + (200*x - 40*x^2 + 2*x^3)*Log[x]^2),x]
Output:
$Aborted
Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {-6+x}{2 \left (x -10\right ) \left (5 x^{2}+\ln \left (x \right )\right )}\) | \(21\) |
default | \(-\frac {-x +6}{2 \left (x -10\right ) \left (5 x^{2}+\ln \left (x \right )\right )}\) | \(23\) |
parallelrisch | \(\frac {-6+x}{10 x^{3}+2 x \ln \left (x \right )-100 x^{2}-20 \ln \left (x \right )}\) | \(27\) |
norman | \(\frac {\frac {x}{2}-3}{5 x^{3}+x \ln \left (x \right )-50 x^{2}-10 \ln \left (x \right )}\) | \(28\) |
Input:
int((-4*x*ln(x)-10*x^4+140*x^3-601*x^2+16*x-60)/((2*x^3-40*x^2+200*x)*ln(x )^2+(20*x^5-400*x^4+2000*x^3)*ln(x)+50*x^7-1000*x^6+5000*x^5),x,method=_RE TURNVERBOSE)
Output:
1/2*(-6+x)/(x-10)/(5*x^2+ln(x))
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-60+16 x-601 x^2+140 x^3-10 x^4-4 x \log (x)}{5000 x^5-1000 x^6+50 x^7+\left (2000 x^3-400 x^4+20 x^5\right ) \log (x)+\left (200 x-40 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {x - 6}{2 \, {\left (5 \, x^{3} - 50 \, x^{2} + {\left (x - 10\right )} \log \left (x\right )\right )}} \] Input:
integrate((-4*x*log(x)-10*x^4+140*x^3-601*x^2+16*x-60)/((2*x^3-40*x^2+200* x)*log(x)^2+(20*x^5-400*x^4+2000*x^3)*log(x)+50*x^7-1000*x^6+5000*x^5),x, algorithm="fricas")
Output:
1/2*(x - 6)/(5*x^3 - 50*x^2 + (x - 10)*log(x))
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-60+16 x-601 x^2+140 x^3-10 x^4-4 x \log (x)}{5000 x^5-1000 x^6+50 x^7+\left (2000 x^3-400 x^4+20 x^5\right ) \log (x)+\left (200 x-40 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {x - 6}{10 x^{3} - 100 x^{2} + \left (2 x - 20\right ) \log {\left (x \right )}} \] Input:
integrate((-4*x*ln(x)-10*x**4+140*x**3-601*x**2+16*x-60)/((2*x**3-40*x**2+ 200*x)*ln(x)**2+(20*x**5-400*x**4+2000*x**3)*ln(x)+50*x**7-1000*x**6+5000* x**5),x)
Output:
(x - 6)/(10*x**3 - 100*x**2 + (2*x - 20)*log(x))
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-60+16 x-601 x^2+140 x^3-10 x^4-4 x \log (x)}{5000 x^5-1000 x^6+50 x^7+\left (2000 x^3-400 x^4+20 x^5\right ) \log (x)+\left (200 x-40 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {x - 6}{2 \, {\left (5 \, x^{3} - 50 \, x^{2} + {\left (x - 10\right )} \log \left (x\right )\right )}} \] Input:
integrate((-4*x*log(x)-10*x^4+140*x^3-601*x^2+16*x-60)/((2*x^3-40*x^2+200* x)*log(x)^2+(20*x^5-400*x^4+2000*x^3)*log(x)+50*x^7-1000*x^6+5000*x^5),x, algorithm="maxima")
Output:
1/2*(x - 6)/(5*x^3 - 50*x^2 + (x - 10)*log(x))
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-60+16 x-601 x^2+140 x^3-10 x^4-4 x \log (x)}{5000 x^5-1000 x^6+50 x^7+\left (2000 x^3-400 x^4+20 x^5\right ) \log (x)+\left (200 x-40 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {x - 6}{2 \, {\left (5 \, x^{3} - 50 \, x^{2} + x \log \left (x\right ) - 10 \, \log \left (x\right )\right )}} \] Input:
integrate((-4*x*log(x)-10*x^4+140*x^3-601*x^2+16*x-60)/((2*x^3-40*x^2+200* x)*log(x)^2+(20*x^5-400*x^4+2000*x^3)*log(x)+50*x^7-1000*x^6+5000*x^5),x, algorithm="giac")
Output:
1/2*(x - 6)/(5*x^3 - 50*x^2 + x*log(x) - 10*log(x))
Time = 4.55 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-60+16 x-601 x^2+140 x^3-10 x^4-4 x \log (x)}{5000 x^5-1000 x^6+50 x^7+\left (2000 x^3-400 x^4+20 x^5\right ) \log (x)+\left (200 x-40 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {x-6}{2\,\left (\ln \left (x\right )+5\,x^2\right )\,\left (x-10\right )} \] Input:
int(-(4*x*log(x) - 16*x + 601*x^2 - 140*x^3 + 10*x^4 + 60)/(log(x)^2*(200* x - 40*x^2 + 2*x^3) + log(x)*(2000*x^3 - 400*x^4 + 20*x^5) + 5000*x^5 - 10 00*x^6 + 50*x^7),x)
Output:
(x - 6)/(2*(log(x) + 5*x^2)*(x - 10))
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-60+16 x-601 x^2+140 x^3-10 x^4-4 x \log (x)}{5000 x^5-1000 x^6+50 x^7+\left (2000 x^3-400 x^4+20 x^5\right ) \log (x)+\left (200 x-40 x^2+2 x^3\right ) \log ^2(x)} \, dx=\frac {x -6}{2 \,\mathrm {log}\left (x \right ) x -20 \,\mathrm {log}\left (x \right )+10 x^{3}-100 x^{2}} \] Input:
int((-4*x*log(x)-10*x^4+140*x^3-601*x^2+16*x-60)/((2*x^3-40*x^2+200*x)*log (x)^2+(20*x^5-400*x^4+2000*x^3)*log(x)+50*x^7-1000*x^6+5000*x^5),x)
Output:
(x - 6)/(2*(log(x)*x - 10*log(x) + 5*x**3 - 50*x**2))