Integrand size = 107, antiderivative size = 24 \[ \int \frac {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+\left (-45+26 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)}{625-750 x+6325 x^2-6060 x^3+16324 x^4-11712 x^5+2304 x^6+\left (-50+30 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)} \, dx=x+\frac {x}{-5+x-24 x^2+\frac {\log (x)}{5-2 x}} \] Output:
x+x/(x-24*x^2-5+ln(x)/(5-2*x))
Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+\left (-45+26 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)}{625-750 x+6325 x^2-6060 x^3+16324 x^4-11712 x^5+2304 x^6+\left (-50+30 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)} \, dx=x-\frac {x (-5+2 x)}{-25+15 x-122 x^2+48 x^3+\log (x)} \] Input:
Integrate[(495 - 648*x + 6905*x^2 - 6540*x^3 + 16420*x^4 - 11712*x^5 + 230 4*x^6 + (-45 + 26*x - 244*x^2 + 96*x^3)*Log[x] + Log[x]^2)/(625 - 750*x + 6325*x^2 - 6060*x^3 + 16324*x^4 - 11712*x^5 + 2304*x^6 + (-50 + 30*x - 244 *x^2 + 96*x^3)*Log[x] + Log[x]^2),x]
Output:
x - (x*(-5 + 2*x))/(-25 + 15*x - 122*x^2 + 48*x^3 + 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.
\(\displaystyle \int \frac {2304 x^6-11712 x^5+16420 x^4-6540 x^3+6905 x^2+\left (96 x^3-244 x^2+26 x-45\right ) \log (x)-648 x+\log ^2(x)+495}{2304 x^6-11712 x^5+16324 x^4-6060 x^3+6325 x^2+\left (96 x^3-244 x^2+30 x-50\right ) \log (x)-750 x+\log ^2(x)+625} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2304 x^6-11712 x^5+16420 x^4-6540 x^3+6905 x^2+\left (96 x^3-244 x^2+26 x-45\right ) \log (x)-648 x+\log ^2(x)+495}{\left (-48 x^3+122 x^2-15 x-\log (x)+25\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {5-4 x}{48 x^3-122 x^2+15 x+\log (x)-25}+\frac {288 x^4-1208 x^3+1250 x^2-73 x-5}{\left (48 x^3-122 x^2+15 x+\log (x)-25\right )^2}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -5 \int \frac {1}{\left (48 x^3-122 x^2+15 x+\log (x)-25\right )^2}dx-73 \int \frac {x}{\left (48 x^3-122 x^2+15 x+\log (x)-25\right )^2}dx+1250 \int \frac {x^2}{\left (48 x^3-122 x^2+15 x+\log (x)-25\right )^2}dx-1208 \int \frac {x^3}{\left (48 x^3-122 x^2+15 x+\log (x)-25\right )^2}dx+5 \int \frac {1}{48 x^3-122 x^2+15 x+\log (x)-25}dx-4 \int \frac {x}{48 x^3-122 x^2+15 x+\log (x)-25}dx+288 \int \frac {x^4}{\left (48 x^3-122 x^2+15 x+\log (x)-25\right )^2}dx+x\) |
Input:
Int[(495 - 648*x + 6905*x^2 - 6540*x^3 + 16420*x^4 - 11712*x^5 + 2304*x^6 + (-45 + 26*x - 244*x^2 + 96*x^3)*Log[x] + Log[x]^2)/(625 - 750*x + 6325*x ^2 - 6060*x^3 + 16324*x^4 - 11712*x^5 + 2304*x^6 + (-50 + 30*x - 244*x^2 + 96*x^3)*Log[x] + Log[x]^2),x]
Output:
$Aborted
Time = 1.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25
method | result | size |
risch | \(x -\frac {\left (-5+2 x \right ) x}{48 x^{3}-122 x^{2}+\ln \left (x \right )+15 x -25}\) | \(30\) |
default | \(\frac {-122 x^{3}-20 x +13 x^{2}+x \ln \left (x \right )+48 x^{4}}{48 x^{3}-122 x^{2}+\ln \left (x \right )+15 x -25}\) | \(44\) |
norman | \(\frac {x \ln \left (x \right )-\frac {3565 x^{2}}{12}+\frac {61 \ln \left (x \right )}{24}+\frac {145 x}{8}+48 x^{4}-\frac {1525}{24}}{48 x^{3}-122 x^{2}+\ln \left (x \right )+15 x -25}\) | \(44\) |
parallelrisch | \(\frac {-122 x^{3}-20 x +13 x^{2}+x \ln \left (x \right )+48 x^{4}}{48 x^{3}-122 x^{2}+\ln \left (x \right )+15 x -25}\) | \(44\) |
Input:
int((ln(x)^2+(96*x^3-244*x^2+26*x-45)*ln(x)+2304*x^6-11712*x^5+16420*x^4-6 540*x^3+6905*x^2-648*x+495)/(ln(x)^2+(96*x^3-244*x^2+30*x-50)*ln(x)+2304*x ^6-11712*x^5+16324*x^4-6060*x^3+6325*x^2-750*x+625),x,method=_RETURNVERBOS E)
Output:
x-(-5+2*x)*x/(48*x^3-122*x^2+ln(x)+15*x-25)
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+\left (-45+26 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)}{625-750 x+6325 x^2-6060 x^3+16324 x^4-11712 x^5+2304 x^6+\left (-50+30 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)} \, dx=\frac {48 \, x^{4} - 122 \, x^{3} + 13 \, x^{2} + x \log \left (x\right ) - 20 \, x}{48 \, x^{3} - 122 \, x^{2} + 15 \, x + \log \left (x\right ) - 25} \] Input:
integrate((log(x)^2+(96*x^3-244*x^2+26*x-45)*log(x)+2304*x^6-11712*x^5+164 20*x^4-6540*x^3+6905*x^2-648*x+495)/(log(x)^2+(96*x^3-244*x^2+30*x-50)*log (x)+2304*x^6-11712*x^5+16324*x^4-6060*x^3+6325*x^2-750*x+625),x, algorithm ="fricas")
Output:
(48*x^4 - 122*x^3 + 13*x^2 + x*log(x) - 20*x)/(48*x^3 - 122*x^2 + 15*x + l og(x) - 25)
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+\left (-45+26 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)}{625-750 x+6325 x^2-6060 x^3+16324 x^4-11712 x^5+2304 x^6+\left (-50+30 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)} \, dx=x + \frac {- 2 x^{2} + 5 x}{48 x^{3} - 122 x^{2} + 15 x + \log {\left (x \right )} - 25} \] Input:
integrate((ln(x)**2+(96*x**3-244*x**2+26*x-45)*ln(x)+2304*x**6-11712*x**5+ 16420*x**4-6540*x**3+6905*x**2-648*x+495)/(ln(x)**2+(96*x**3-244*x**2+30*x -50)*ln(x)+2304*x**6-11712*x**5+16324*x**4-6060*x**3+6325*x**2-750*x+625), x)
Output:
x + (-2*x**2 + 5*x)/(48*x**3 - 122*x**2 + 15*x + log(x) - 25)
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+\left (-45+26 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)}{625-750 x+6325 x^2-6060 x^3+16324 x^4-11712 x^5+2304 x^6+\left (-50+30 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)} \, dx=\frac {48 \, x^{4} - 122 \, x^{3} + 13 \, x^{2} + x \log \left (x\right ) - 20 \, x}{48 \, x^{3} - 122 \, x^{2} + 15 \, x + \log \left (x\right ) - 25} \] Input:
integrate((log(x)^2+(96*x^3-244*x^2+26*x-45)*log(x)+2304*x^6-11712*x^5+164 20*x^4-6540*x^3+6905*x^2-648*x+495)/(log(x)^2+(96*x^3-244*x^2+30*x-50)*log (x)+2304*x^6-11712*x^5+16324*x^4-6060*x^3+6325*x^2-750*x+625),x, algorithm ="maxima")
Output:
(48*x^4 - 122*x^3 + 13*x^2 + x*log(x) - 20*x)/(48*x^3 - 122*x^2 + 15*x + l og(x) - 25)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+\left (-45+26 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)}{625-750 x+6325 x^2-6060 x^3+16324 x^4-11712 x^5+2304 x^6+\left (-50+30 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)} \, dx=x - \frac {2 \, x^{2} - 5 \, x}{48 \, x^{3} - 122 \, x^{2} + 15 \, x + \log \left (x\right ) - 25} \] Input:
integrate((log(x)^2+(96*x^3-244*x^2+26*x-45)*log(x)+2304*x^6-11712*x^5+164 20*x^4-6540*x^3+6905*x^2-648*x+495)/(log(x)^2+(96*x^3-244*x^2+30*x-50)*log (x)+2304*x^6-11712*x^5+16324*x^4-6060*x^3+6325*x^2-750*x+625),x, algorithm ="giac")
Output:
x - (2*x^2 - 5*x)/(48*x^3 - 122*x^2 + 15*x + log(x) - 25)
Time = 1.69 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+\left (-45+26 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)}{625-750 x+6325 x^2-6060 x^3+16324 x^4-11712 x^5+2304 x^6+\left (-50+30 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)} \, dx=x+\frac {5\,x-2\,x^2}{15\,x+\ln \left (x\right )-122\,x^2+48\,x^3-25} \] Input:
int((log(x)^2 - 648*x + 6905*x^2 - 6540*x^3 + 16420*x^4 - 11712*x^5 + 2304 *x^6 + log(x)*(26*x - 244*x^2 + 96*x^3 - 45) + 495)/(log(x)^2 - 750*x + 63 25*x^2 - 6060*x^3 + 16324*x^4 - 11712*x^5 + 2304*x^6 + log(x)*(30*x - 244* x^2 + 96*x^3 - 50) + 625),x)
Output:
x + (5*x - 2*x^2)/(15*x + log(x) - 122*x^2 + 48*x^3 - 25)
Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {495-648 x+6905 x^2-6540 x^3+16420 x^4-11712 x^5+2304 x^6+\left (-45+26 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)}{625-750 x+6325 x^2-6060 x^3+16324 x^4-11712 x^5+2304 x^6+\left (-50+30 x-244 x^2+96 x^3\right ) \log (x)+\log ^2(x)} \, dx=\frac {24 \,\mathrm {log}\left (x \right ) x +61 \,\mathrm {log}\left (x \right )+1152 x^{4}-7130 x^{2}+435 x -1525}{24 \,\mathrm {log}\left (x \right )+1152 x^{3}-2928 x^{2}+360 x -600} \] Input:
int((log(x)^2+(96*x^3-244*x^2+26*x-45)*log(x)+2304*x^6-11712*x^5+16420*x^4 -6540*x^3+6905*x^2-648*x+495)/(log(x)^2+(96*x^3-244*x^2+30*x-50)*log(x)+23 04*x^6-11712*x^5+16324*x^4-6060*x^3+6325*x^2-750*x+625),x)
Output:
(24*log(x)*x + 61*log(x) + 1152*x**4 - 7130*x**2 + 435*x - 1525)/(24*(log( x) + 48*x**3 - 122*x**2 + 15*x - 25))