Integrand size = 95, antiderivative size = 23 \[ \int \frac {-25+190 x-81 x^2-56 x^3+\left (-25+70 x+31 x^2\right ) \log (x)+(-10-4 x) \log ^2(x)}{400 x^4+160 x^5+16 x^6+\left (-200 x^3-80 x^4-8 x^5\right ) \log (x)+\left (25 x^2+10 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {-x+\frac {2}{5+x}+\frac {1}{-4 x+\log (x)}}{x} \] Output:
(2/3/(5/3+1/3*x)+4/(4*ln(x)-16*x)-x)/x
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-25+190 x-81 x^2-56 x^3+\left (-25+70 x+31 x^2\right ) \log (x)+(-10-4 x) \log ^2(x)}{400 x^4+160 x^5+16 x^6+\left (-200 x^3-80 x^4-8 x^5\right ) \log (x)+\left (25 x^2+10 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {2}{5 x}-\frac {2}{5 (5+x)}+\frac {1}{x (-4 x+\log (x))} \] Input:
Integrate[(-25 + 190*x - 81*x^2 - 56*x^3 + (-25 + 70*x + 31*x^2)*Log[x] + (-10 - 4*x)*Log[x]^2)/(400*x^4 + 160*x^5 + 16*x^6 + (-200*x^3 - 80*x^4 - 8 *x^5)*Log[x] + (25*x^2 + 10*x^3 + x^4)*Log[x]^2),x]
Output:
2/(5*x) - 2/(5*(5 + x)) + 1/(x*(-4*x + 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.
\(\displaystyle \int \frac {-56 x^3-81 x^2+\left (31 x^2+70 x-25\right ) \log (x)+190 x+(-4 x-10) \log ^2(x)-25}{16 x^6+160 x^5+400 x^4+\left (-8 x^5-80 x^4-200 x^3\right ) \log (x)+\left (x^4+10 x^3+25 x^2\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-56 x^3-81 x^2+\left (31 x^2+70 x-25\right ) \log (x)+190 x+(-4 x-10) \log ^2(x)-25}{x^2 (x+5)^2 (4 x-\log (x))^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 (2 x+5)}{x^2 (x+5)^2}+\frac {1}{x^2 (4 x-\log (x))}+\frac {4 x-1}{x^2 (4 x-\log (x))^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {1}{x^2 (4 x-\log (x))^2}dx+\int \frac {1}{x^2 (4 x-\log (x))}dx+4 \int \frac {1}{x (4 x-\log (x))^2}dx+\frac {2}{x (x+5)}\) |
Input:
Int[(-25 + 190*x - 81*x^2 - 56*x^3 + (-25 + 70*x + 31*x^2)*Log[x] + (-10 - 4*x)*Log[x]^2)/(400*x^4 + 160*x^5 + 16*x^6 + (-200*x^3 - 80*x^4 - 8*x^5)* Log[x] + (25*x^2 + 10*x^3 + x^4)*Log[x]^2),x]
Output:
$Aborted
Time = 0.71 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\frac {2}{\left (5+x \right ) x}-\frac {1}{x \left (4 x -\ln \left (x \right )\right )}\) | \(27\) |
default | \(-\frac {-5-2 \ln \left (x \right )+7 x}{x \left (5+x \right ) \left (\ln \left (x \right )-4 x \right )}\) | \(28\) |
norman | \(\frac {-5-2 \ln \left (x \right )+7 x}{x \left (4 x^{2}-x \ln \left (x \right )+20 x -5 \ln \left (x \right )\right )}\) | \(34\) |
parallelrisch | \(\frac {-5-2 \ln \left (x \right )+7 x}{x \left (4 x^{2}-x \ln \left (x \right )+20 x -5 \ln \left (x \right )\right )}\) | \(34\) |
Input:
int(((-4*x-10)*ln(x)^2+(31*x^2+70*x-25)*ln(x)-56*x^3-81*x^2+190*x-25)/((x^ 4+10*x^3+25*x^2)*ln(x)^2+(-8*x^5-80*x^4-200*x^3)*ln(x)+16*x^6+160*x^5+400* x^4),x,method=_RETURNVERBOSE)
Output:
2/(5+x)/x-1/x/(4*x-ln(x))
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {-25+190 x-81 x^2-56 x^3+\left (-25+70 x+31 x^2\right ) \log (x)+(-10-4 x) \log ^2(x)}{400 x^4+160 x^5+16 x^6+\left (-200 x^3-80 x^4-8 x^5\right ) \log (x)+\left (25 x^2+10 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {7 \, x - 2 \, \log \left (x\right ) - 5}{4 \, x^{3} + 20 \, x^{2} - {\left (x^{2} + 5 \, x\right )} \log \left (x\right )} \] Input:
integrate(((-4*x-10)*log(x)^2+(31*x^2+70*x-25)*log(x)-56*x^3-81*x^2+190*x- 25)/((x^4+10*x^3+25*x^2)*log(x)^2+(-8*x^5-80*x^4-200*x^3)*log(x)+16*x^6+16 0*x^5+400*x^4),x, algorithm="fricas")
Output:
(7*x - 2*log(x) - 5)/(4*x^3 + 20*x^2 - (x^2 + 5*x)*log(x))
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-25+190 x-81 x^2-56 x^3+\left (-25+70 x+31 x^2\right ) \log (x)+(-10-4 x) \log ^2(x)}{400 x^4+160 x^5+16 x^6+\left (-200 x^3-80 x^4-8 x^5\right ) \log (x)+\left (25 x^2+10 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {2}{x^{2} + 5 x} + \frac {1}{- 4 x^{2} + x \log {\left (x \right )}} \] Input:
integrate(((-4*x-10)*ln(x)**2+(31*x**2+70*x-25)*ln(x)-56*x**3-81*x**2+190* x-25)/((x**4+10*x**3+25*x**2)*ln(x)**2+(-8*x**5-80*x**4-200*x**3)*ln(x)+16 *x**6+160*x**5+400*x**4),x)
Output:
2/(x**2 + 5*x) + 1/(-4*x**2 + x*log(x))
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {-25+190 x-81 x^2-56 x^3+\left (-25+70 x+31 x^2\right ) \log (x)+(-10-4 x) \log ^2(x)}{400 x^4+160 x^5+16 x^6+\left (-200 x^3-80 x^4-8 x^5\right ) \log (x)+\left (25 x^2+10 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {7 \, x - 2 \, \log \left (x\right ) - 5}{4 \, x^{3} + 20 \, x^{2} - {\left (x^{2} + 5 \, x\right )} \log \left (x\right )} \] Input:
integrate(((-4*x-10)*log(x)^2+(31*x^2+70*x-25)*log(x)-56*x^3-81*x^2+190*x- 25)/((x^4+10*x^3+25*x^2)*log(x)^2+(-8*x^5-80*x^4-200*x^3)*log(x)+16*x^6+16 0*x^5+400*x^4),x, algorithm="maxima")
Output:
(7*x - 2*log(x) - 5)/(4*x^3 + 20*x^2 - (x^2 + 5*x)*log(x))
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {-25+190 x-81 x^2-56 x^3+\left (-25+70 x+31 x^2\right ) \log (x)+(-10-4 x) \log ^2(x)}{400 x^4+160 x^5+16 x^6+\left (-200 x^3-80 x^4-8 x^5\right ) \log (x)+\left (25 x^2+10 x^3+x^4\right ) \log ^2(x)} \, dx=-\frac {1}{4 \, x^{2} - x \log \left (x\right )} - \frac {2}{5 \, {\left (x + 5\right )}} + \frac {2}{5 \, x} \] Input:
integrate(((-4*x-10)*log(x)^2+(31*x^2+70*x-25)*log(x)-56*x^3-81*x^2+190*x- 25)/((x^4+10*x^3+25*x^2)*log(x)^2+(-8*x^5-80*x^4-200*x^3)*log(x)+16*x^6+16 0*x^5+400*x^4),x, algorithm="giac")
Output:
-1/(4*x^2 - x*log(x)) - 2/5/(x + 5) + 2/5/x
Time = 2.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-25+190 x-81 x^2-56 x^3+\left (-25+70 x+31 x^2\right ) \log (x)+(-10-4 x) \log ^2(x)}{400 x^4+160 x^5+16 x^6+\left (-200 x^3-80 x^4-8 x^5\right ) \log (x)+\left (25 x^2+10 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {2}{x\,\left (x+5\right )}-\frac {1}{x\,\left (4\,x-\ln \left (x\right )\right )} \] Input:
int(-(81*x^2 - log(x)*(70*x + 31*x^2 - 25) - 190*x + 56*x^3 + log(x)^2*(4* x + 10) + 25)/(log(x)^2*(25*x^2 + 10*x^3 + x^4) - log(x)*(200*x^3 + 80*x^4 + 8*x^5) + 400*x^4 + 160*x^5 + 16*x^6),x)
Output:
2/(x*(x + 5)) - 1/(x*(4*x - log(x)))
Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-25+190 x-81 x^2-56 x^3+\left (-25+70 x+31 x^2\right ) \log (x)+(-10-4 x) \log ^2(x)}{400 x^4+160 x^5+16 x^6+\left (-200 x^3-80 x^4-8 x^5\right ) \log (x)+\left (25 x^2+10 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {2 \,\mathrm {log}\left (x \right )-7 x +5}{x \left (\mathrm {log}\left (x \right ) x +5 \,\mathrm {log}\left (x \right )-4 x^{2}-20 x \right )} \] Input:
int(((-4*x-10)*log(x)^2+(31*x^2+70*x-25)*log(x)-56*x^3-81*x^2+190*x-25)/(( x^4+10*x^3+25*x^2)*log(x)^2+(-8*x^5-80*x^4-200*x^3)*log(x)+16*x^6+160*x^5+ 400*x^4),x)
Output:
(2*log(x) - 7*x + 5)/(x*(log(x)*x + 5*log(x) - 4*x**2 - 20*x))