Integrand size = 99, antiderivative size = 33 \[ \int \frac {-200-280 x+18 x^2+122 x^3+34 x^4+\left (20 x^2-2 x^4\right ) \log (x)}{100-140 x+9 x^2+68 x^3-24 x^4-8 x^5+4 x^6+\left (20 x^2-14 x^3-4 x^4+4 x^5\right ) \log (x)+x^4 \log ^2(x)} \, dx=-3+\frac {5+x}{3-x-\frac {5+\frac {x^2 \log (x)}{2+x}}{2 x}} \]
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {-200-280 x+18 x^2+122 x^3+34 x^4+\left (20 x^2-2 x^4\right ) \log (x)}{100-140 x+9 x^2+68 x^3-24 x^4-8 x^5+4 x^6+\left (20 x^2-14 x^3-4 x^4+4 x^5\right ) \log (x)+x^4 \log ^2(x)} \, dx=-\frac {2 x \left (10+7 x+x^2\right )}{10-7 x-2 x^2+2 x^3+x^2 \log (x)} \]
Integrate[(-200 - 280*x + 18*x^2 + 122*x^3 + 34*x^4 + (20*x^2 - 2*x^4)*Log [x])/(100 - 140*x + 9*x^2 + 68*x^3 - 24*x^4 - 8*x^5 + 4*x^6 + (20*x^2 - 14 *x^3 - 4*x^4 + 4*x^5)*Log[x] + x^4*Log[x]^2),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.
\(\displaystyle \int \frac {34 x^4+122 x^3+18 x^2+\left (20 x^2-2 x^4\right ) \log (x)-280 x-200}{4 x^6-8 x^5-24 x^4+x^4 \log ^2(x)+68 x^3+9 x^2+\left (4 x^5-4 x^4-14 x^3+20 x^2\right ) \log (x)-140 x+100} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 \left (17 x^4+61 x^3+9 x^2-\left (x^2-10\right ) x^2 \log (x)-140 x-100\right )}{\left (2 x^3-2 x^2+x^2 \log (x)-7 x+10\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {-17 x^4-61 x^3-\left (10-x^2\right ) \log (x) x^2-9 x^2+140 x+100}{\left (2 x^3+\log (x) x^2-2 x^2-7 x+10\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {-17 x^4-61 x^3-\left (10-x^2\right ) \log (x) x^2-9 x^2+140 x+100}{\left (2 x^3+\log (x) x^2-2 x^2-7 x+10\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {x^2-10}{2 x^3+\log (x) x^2-2 x^2-7 x+10}+\frac {-2 x^5-15 x^4-34 x^3-39 x^2+70 x+200}{\left (2 x^3+\log (x) x^2-2 x^2-7 x+10\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (200 \int \frac {1}{\left (2 x^3+\log (x) x^2-2 x^2-7 x+10\right )^2}dx+70 \int \frac {x}{\left (2 x^3+\log (x) x^2-2 x^2-7 x+10\right )^2}dx-39 \int \frac {x^2}{\left (2 x^3+\log (x) x^2-2 x^2-7 x+10\right )^2}dx-34 \int \frac {x^3}{\left (2 x^3+\log (x) x^2-2 x^2-7 x+10\right )^2}dx-10 \int \frac {1}{2 x^3+\log (x) x^2-2 x^2-7 x+10}dx+\int \frac {x^2}{2 x^3+\log (x) x^2-2 x^2-7 x+10}dx-2 \int \frac {x^5}{\left (2 x^3+\log (x) x^2-2 x^2-7 x+10\right )^2}dx-15 \int \frac {x^4}{\left (2 x^3+\log (x) x^2-2 x^2-7 x+10\right )^2}dx\right )\) |
Int[(-200 - 280*x + 18*x^2 + 122*x^3 + 34*x^4 + (20*x^2 - 2*x^4)*Log[x])/( 100 - 140*x + 9*x^2 + 68*x^3 - 24*x^4 - 8*x^5 + 4*x^6 + (20*x^2 - 14*x^3 - 4*x^4 + 4*x^5)*Log[x] + x^4*Log[x]^2),x]
3.22.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.69 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06
method | result | size |
risch | \(-\frac {2 \left (x^{2}+7 x +10\right ) x}{x^{2} \ln \left (x \right )+2 x^{3}-2 x^{2}-7 x +10}\) | \(35\) |
default | \(-\frac {2 \left (x^{3}+7 x^{2}+10 x \right )}{x^{2} \ln \left (x \right )+2 x^{3}-2 x^{2}-7 x +10}\) | \(38\) |
parallelrisch | \(\frac {-2 x^{3}-14 x^{2}-20 x}{x^{2} \ln \left (x \right )+2 x^{3}-2 x^{2}-7 x +10}\) | \(39\) |
norman | \(\frac {-27 x -16 x^{2}+x^{2} \ln \left (x \right )+10}{x^{2} \ln \left (x \right )+2 x^{3}-2 x^{2}-7 x +10}\) | \(41\) |
int(((-2*x^4+20*x^2)*ln(x)+34*x^4+122*x^3+18*x^2-280*x-200)/(x^4*ln(x)^2+( 4*x^5-4*x^4-14*x^3+20*x^2)*ln(x)+4*x^6-8*x^5-24*x^4+68*x^3+9*x^2-140*x+100 ),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-200-280 x+18 x^2+122 x^3+34 x^4+\left (20 x^2-2 x^4\right ) \log (x)}{100-140 x+9 x^2+68 x^3-24 x^4-8 x^5+4 x^6+\left (20 x^2-14 x^3-4 x^4+4 x^5\right ) \log (x)+x^4 \log ^2(x)} \, dx=-\frac {2 \, {\left (x^{3} + 7 \, x^{2} + 10 \, x\right )}}{2 \, x^{3} + x^{2} \log \left (x\right ) - 2 \, x^{2} - 7 \, x + 10} \]
integrate(((-2*x^4+20*x^2)*log(x)+34*x^4+122*x^3+18*x^2-280*x-200)/(x^4*lo g(x)^2+(4*x^5-4*x^4-14*x^3+20*x^2)*log(x)+4*x^6-8*x^5-24*x^4+68*x^3+9*x^2- 140*x+100),x, algorithm=\
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {-200-280 x+18 x^2+122 x^3+34 x^4+\left (20 x^2-2 x^4\right ) \log (x)}{100-140 x+9 x^2+68 x^3-24 x^4-8 x^5+4 x^6+\left (20 x^2-14 x^3-4 x^4+4 x^5\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {- 2 x^{3} - 14 x^{2} - 20 x}{2 x^{3} + x^{2} \log {\left (x \right )} - 2 x^{2} - 7 x + 10} \]
integrate(((-2*x**4+20*x**2)*ln(x)+34*x**4+122*x**3+18*x**2-280*x-200)/(x* *4*ln(x)**2+(4*x**5-4*x**4-14*x**3+20*x**2)*ln(x)+4*x**6-8*x**5-24*x**4+68 *x**3+9*x**2-140*x+100),x)
Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-200-280 x+18 x^2+122 x^3+34 x^4+\left (20 x^2-2 x^4\right ) \log (x)}{100-140 x+9 x^2+68 x^3-24 x^4-8 x^5+4 x^6+\left (20 x^2-14 x^3-4 x^4+4 x^5\right ) \log (x)+x^4 \log ^2(x)} \, dx=-\frac {2 \, {\left (x^{3} + 7 \, x^{2} + 10 \, x\right )}}{2 \, x^{3} + x^{2} \log \left (x\right ) - 2 \, x^{2} - 7 \, x + 10} \]
integrate(((-2*x^4+20*x^2)*log(x)+34*x^4+122*x^3+18*x^2-280*x-200)/(x^4*lo g(x)^2+(4*x^5-4*x^4-14*x^3+20*x^2)*log(x)+4*x^6-8*x^5-24*x^4+68*x^3+9*x^2- 140*x+100),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-200-280 x+18 x^2+122 x^3+34 x^4+\left (20 x^2-2 x^4\right ) \log (x)}{100-140 x+9 x^2+68 x^3-24 x^4-8 x^5+4 x^6+\left (20 x^2-14 x^3-4 x^4+4 x^5\right ) \log (x)+x^4 \log ^2(x)} \, dx=-\frac {2 \, {\left (x^{3} + 7 \, x^{2} + 10 \, x\right )}}{2 \, x^{3} + x^{2} \log \left (x\right ) - 2 \, x^{2} - 7 \, x + 10} \]
integrate(((-2*x^4+20*x^2)*log(x)+34*x^4+122*x^3+18*x^2-280*x-200)/(x^4*lo g(x)^2+(4*x^5-4*x^4-14*x^3+20*x^2)*log(x)+4*x^6-8*x^5-24*x^4+68*x^3+9*x^2- 140*x+100),x, algorithm=\
Time = 13.91 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-200-280 x+18 x^2+122 x^3+34 x^4+\left (20 x^2-2 x^4\right ) \log (x)}{100-140 x+9 x^2+68 x^3-24 x^4-8 x^5+4 x^6+\left (20 x^2-14 x^3-4 x^4+4 x^5\right ) \log (x)+x^4 \log ^2(x)} \, dx=-\frac {2\,x^3+14\,x^2+20\,x}{x^2\,\ln \left (x\right )-7\,x-2\,x^2+2\,x^3+10} \]