Integrand size = 122, antiderivative size = 33 \begin {dmath*} \int \frac {8 x^2+9 x^3+6 x^4+2 x^5+\left (120 x+4 x^2-50 x^3-29 x^4-4 x^5\right ) \log (x)+\left (-300-120 x+88 x^2+90 x^3+24 x^4+2 x^5\right ) \log ^2(x)}{2 x^4+\left (-20 x^3-4 x^4\right ) \log (x)+\left (50 x^2+20 x^3+2 x^4\right ) \log ^2(x)} \, dx=\frac {(4+x) \left (3-x \left (-x+\frac {x}{-x+(5+x) \log (x)}\right )\right )}{2 x} \end {dmath*}
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \begin {dmath*} \int \frac {8 x^2+9 x^3+6 x^4+2 x^5+\left (120 x+4 x^2-50 x^3-29 x^4-4 x^5\right ) \log (x)+\left (-300-120 x+88 x^2+90 x^3+24 x^4+2 x^5\right ) \log ^2(x)}{2 x^4+\left (-20 x^3-4 x^4\right ) \log (x)+\left (50 x^2+20 x^3+2 x^4\right ) \log ^2(x)} \, dx=\frac {1}{2} \left (\frac {12}{x}+4 x+x^2-\frac {x (4+x)}{-x+5 \log (x)+x \log (x)}\right ) \end {dmath*}
Integrate[(8*x^2 + 9*x^3 + 6*x^4 + 2*x^5 + (120*x + 4*x^2 - 50*x^3 - 29*x^ 4 - 4*x^5)*Log[x] + (-300 - 120*x + 88*x^2 + 90*x^3 + 24*x^4 + 2*x^5)*Log[ x]^2)/(2*x^4 + (-20*x^3 - 4*x^4)*Log[x] + (50*x^2 + 20*x^3 + 2*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.
\(\displaystyle \int \frac {2 x^5+6 x^4+9 x^3+8 x^2+\left (2 x^5+24 x^4+90 x^3+88 x^2-120 x-300\right ) \log ^2(x)+\left (-4 x^5-29 x^4-50 x^3+4 x^2+120 x\right ) \log (x)}{2 x^4+\left (-4 x^4-20 x^3\right ) \log (x)+\left (2 x^4+20 x^3+50 x^2\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^5+6 x^4+9 x^3+8 x^2+\left (2 x^5+24 x^4+90 x^3+88 x^2-120 x-300\right ) \log ^2(x)+\left (-4 x^5-29 x^4-50 x^3+4 x^2+120 x\right ) \log (x)}{2 x^2 (x+x (-\log (x))-5 \log (x))^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {2 x^5+6 x^4+9 x^3+8 x^2-2 \left (-x^5-12 x^4-45 x^3-44 x^2+60 x+150\right ) \log ^2(x)+\left (-4 x^5-29 x^4-50 x^3+4 x^2+120 x\right ) \log (x)}{x^2 (-\log (x) x+x-5 \log (x))^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (\frac {-x^2-10 x-20}{(x+5) (\log (x) x-x+5 \log (x))}+\frac {2 \left (x^3+2 x^2-6\right )}{x^2}+\frac {x^3+9 x^2+45 x+100}{(x+5) (\log (x) x-x+5 \log (x))^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\int \frac {x^2}{(\log (x) x-x+5 \log (x))^2}dx+25 \int \frac {1}{(\log (x) x-x+5 \log (x))^2}dx+4 \int \frac {x}{(\log (x) x-x+5 \log (x))^2}dx-25 \int \frac {1}{(x+5) (\log (x) x-x+5 \log (x))^2}dx-5 \int \frac {1}{\log (x) x-x+5 \log (x)}dx-\int \frac {x}{\log (x) x-x+5 \log (x)}dx+5 \int \frac {1}{(x+5) (\log (x) x-x+5 \log (x))}dx+x^2+4 x+\frac {12}{x}\right )\) |
Int[(8*x^2 + 9*x^3 + 6*x^4 + 2*x^5 + (120*x + 4*x^2 - 50*x^3 - 29*x^4 - 4* x^5)*Log[x] + (-300 - 120*x + 88*x^2 + 90*x^3 + 24*x^4 + 2*x^5)*Log[x]^2)/ (2*x^4 + (-20*x^3 - 4*x^4)*Log[x] + (50*x^2 + 20*x^3 + 2*x^4)*Log[x]^2),x]
3.5.28.3.1 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]]
Time = 1.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {x^{3}+4 x^{2}+12}{2 x}-\frac {\left (4+x \right ) x}{2 \left (x \ln \left (x \right )+5 \ln \left (x \right )-x \right )}\) | \(37\) |
default | \(\frac {16 x^{2}+x^{4} \ln \left (x \right )-88 x \ln \left (x \right )+60 \ln \left (x \right )-5 x^{3}-x^{4}+9 x^{3} \ln \left (x \right )-12 x}{2 x \left (x \ln \left (x \right )+5 \ln \left (x \right )-x \right )}\) | \(61\) |
norman | \(\frac {8 x^{2}-44 x \ln \left (x \right )-6 x -\frac {5 x^{3}}{2}-\frac {x^{4}}{2}+\frac {9 x^{3} \ln \left (x \right )}{2}+\frac {x^{4} \ln \left (x \right )}{2}+30 \ln \left (x \right )}{x \left (x \ln \left (x \right )+5 \ln \left (x \right )-x \right )}\) | \(61\) |
parallelrisch | \(\frac {16 x^{2}+x^{4} \ln \left (x \right )-88 x \ln \left (x \right )+60 \ln \left (x \right )-5 x^{3}-x^{4}+9 x^{3} \ln \left (x \right )-12 x}{2 x \left (x \ln \left (x \right )+5 \ln \left (x \right )-x \right )}\) | \(61\) |
int(((2*x^5+24*x^4+90*x^3+88*x^2-120*x-300)*ln(x)^2+(-4*x^5-29*x^4-50*x^3+ 4*x^2+120*x)*ln(x)+2*x^5+6*x^4+9*x^3+8*x^2)/((2*x^4+20*x^3+50*x^2)*ln(x)^2 +(-4*x^4-20*x^3)*ln(x)+2*x^4),x,method=_RETURNVERBOSE)
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \begin {dmath*} \int \frac {8 x^2+9 x^3+6 x^4+2 x^5+\left (120 x+4 x^2-50 x^3-29 x^4-4 x^5\right ) \log (x)+\left (-300-120 x+88 x^2+90 x^3+24 x^4+2 x^5\right ) \log ^2(x)}{2 x^4+\left (-20 x^3-4 x^4\right ) \log (x)+\left (50 x^2+20 x^3+2 x^4\right ) \log ^2(x)} \, dx=\frac {x^{4} + 5 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 9 \, x^{3} + 20 \, x^{2} + 12 \, x + 60\right )} \log \left (x\right ) + 12 \, x}{2 \, {\left (x^{2} - {\left (x^{2} + 5 \, x\right )} \log \left (x\right )\right )}} \end {dmath*}
integrate(((2*x^5+24*x^4+90*x^3+88*x^2-120*x-300)*log(x)^2+(-4*x^5-29*x^4- 50*x^3+4*x^2+120*x)*log(x)+2*x^5+6*x^4+9*x^3+8*x^2)/((2*x^4+20*x^3+50*x^2) *log(x)^2+(-4*x^4-20*x^3)*log(x)+2*x^4),x, algorithm=\
1/2*(x^4 + 5*x^3 + 4*x^2 - (x^4 + 9*x^3 + 20*x^2 + 12*x + 60)*log(x) + 12* x)/(x^2 - (x^2 + 5*x)*log(x))
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \begin {dmath*} \int \frac {8 x^2+9 x^3+6 x^4+2 x^5+\left (120 x+4 x^2-50 x^3-29 x^4-4 x^5\right ) \log (x)+\left (-300-120 x+88 x^2+90 x^3+24 x^4+2 x^5\right ) \log ^2(x)}{2 x^4+\left (-20 x^3-4 x^4\right ) \log (x)+\left (50 x^2+20 x^3+2 x^4\right ) \log ^2(x)} \, dx=\frac {x^{2}}{2} + 2 x + \frac {- x^{2} - 4 x}{- 2 x + \left (2 x + 10\right ) \log {\left (x \right )}} + \frac {6}{x} \end {dmath*}
integrate(((2*x**5+24*x**4+90*x**3+88*x**2-120*x-300)*ln(x)**2+(-4*x**5-29 *x**4-50*x**3+4*x**2+120*x)*ln(x)+2*x**5+6*x**4+9*x**3+8*x**2)/((2*x**4+20 *x**3+50*x**2)*ln(x)**2+(-4*x**4-20*x**3)*ln(x)+2*x**4),x)
Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \begin {dmath*} \int \frac {8 x^2+9 x^3+6 x^4+2 x^5+\left (120 x+4 x^2-50 x^3-29 x^4-4 x^5\right ) \log (x)+\left (-300-120 x+88 x^2+90 x^3+24 x^4+2 x^5\right ) \log ^2(x)}{2 x^4+\left (-20 x^3-4 x^4\right ) \log (x)+\left (50 x^2+20 x^3+2 x^4\right ) \log ^2(x)} \, dx=\frac {x^{4} + 5 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 9 \, x^{3} + 20 \, x^{2} + 12 \, x + 60\right )} \log \left (x\right ) + 12 \, x}{2 \, {\left (x^{2} - {\left (x^{2} + 5 \, x\right )} \log \left (x\right )\right )}} \end {dmath*}
integrate(((2*x^5+24*x^4+90*x^3+88*x^2-120*x-300)*log(x)^2+(-4*x^5-29*x^4- 50*x^3+4*x^2+120*x)*log(x)+2*x^5+6*x^4+9*x^3+8*x^2)/((2*x^4+20*x^3+50*x^2) *log(x)^2+(-4*x^4-20*x^3)*log(x)+2*x^4),x, algorithm=\
1/2*(x^4 + 5*x^3 + 4*x^2 - (x^4 + 9*x^3 + 20*x^2 + 12*x + 60)*log(x) + 12* x)/(x^2 - (x^2 + 5*x)*log(x))
Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \begin {dmath*} \int \frac {8 x^2+9 x^3+6 x^4+2 x^5+\left (120 x+4 x^2-50 x^3-29 x^4-4 x^5\right ) \log (x)+\left (-300-120 x+88 x^2+90 x^3+24 x^4+2 x^5\right ) \log ^2(x)}{2 x^4+\left (-20 x^3-4 x^4\right ) \log (x)+\left (50 x^2+20 x^3+2 x^4\right ) \log ^2(x)} \, dx=\frac {1}{2} \, x^{2} + 2 \, x - \frac {x^{2} + 4 \, x}{2 \, {\left (x \log \left (x\right ) - x + 5 \, \log \left (x\right )\right )}} + \frac {6}{x} \end {dmath*}
integrate(((2*x^5+24*x^4+90*x^3+88*x^2-120*x-300)*log(x)^2+(-4*x^5-29*x^4- 50*x^3+4*x^2+120*x)*log(x)+2*x^5+6*x^4+9*x^3+8*x^2)/((2*x^4+20*x^3+50*x^2) *log(x)^2+(-4*x^4-20*x^3)*log(x)+2*x^4),x, algorithm=\
Time = 14.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82 \begin {dmath*} \int \frac {8 x^2+9 x^3+6 x^4+2 x^5+\left (120 x+4 x^2-50 x^3-29 x^4-4 x^5\right ) \log (x)+\left (-300-120 x+88 x^2+90 x^3+24 x^4+2 x^5\right ) \log ^2(x)}{2 x^4+\left (-20 x^3-4 x^4\right ) \log (x)+\left (50 x^2+20 x^3+2 x^4\right ) \log ^2(x)} \, dx=2\,x+\frac {6}{x}+\frac {x^2}{2}+\frac {x^5+9\,x^4+45\,x^3+100\,x^2}{2\,\left (x-\ln \left (x\right )\,\left (x+5\right )\right )\,\left (x^3+5\,x^2+25\,x\right )} \end {dmath*}
int((log(x)^2*(88*x^2 - 120*x + 90*x^3 + 24*x^4 + 2*x^5 - 300) - log(x)*(5 0*x^3 - 4*x^2 - 120*x + 29*x^4 + 4*x^5) + 8*x^2 + 9*x^3 + 6*x^4 + 2*x^5)/( log(x)^2*(50*x^2 + 20*x^3 + 2*x^4) - log(x)*(20*x^3 + 4*x^4) + 2*x^4),x)