Integrand size = 61, antiderivative size = 22 \begin {dmath*} \int \frac {45 x^4-27 x^5+\left (180 x^3-144 x^4+27 x^5\right ) \log (-2+x)}{\left (2000-4600 x+3960 x^2-1512 x^3+216 x^4\right ) \log ^3(-2+x)} \, dx=15+\frac {x^4}{16 \left (-\frac {5}{3}+x\right )^2 \log ^2(-2+x)} \end {dmath*}
Time = 0.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {45 x^4-27 x^5+\left (180 x^3-144 x^4+27 x^5\right ) \log (-2+x)}{\left (2000-4600 x+3960 x^2-1512 x^3+216 x^4\right ) \log ^3(-2+x)} \, dx=\frac {9 x^4}{16 (1+3 (-2+x))^2 \log ^2(-2+x)} \end {dmath*}
Integrate[(45*x^4 - 27*x^5 + (180*x^3 - 144*x^4 + 27*x^5)*Log[-2 + x])/((2 000 - 4600*x + 3960*x^2 - 1512*x^3 + 216*x^4)*Log[-2 + x]^3),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.
\(\displaystyle \int \frac {-27 x^5+45 x^4+\left (27 x^5-144 x^4+180 x^3\right ) \log (x-2)}{\left (216 x^4-1512 x^3+3960 x^2-4600 x+2000\right ) \log ^3(x-2)} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {-27 x^5+45 x^4+\left (27 x^5-144 x^4+180 x^3\right ) \log (x-2)}{8 (x-2) \log ^3(x-2)}-\frac {3 \left (-27 x^5+45 x^4+\left (27 x^5-144 x^4+180 x^3\right ) \log (x-2)\right )}{8 (3 x-5) \log ^3(x-2)}-\frac {3 \left (-27 x^5+45 x^4+\left (27 x^5-144 x^4+180 x^3\right ) \log (x-2)\right )}{8 (3 x-5)^2 \log ^3(x-2)}-\frac {3 \left (-27 x^5+45 x^4+\left (27 x^5-144 x^4+180 x^3\right ) \log (x-2)\right )}{8 (3 x-5)^3 \log ^3(x-2)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {625}{24} \int \frac {1}{(3 x-5)^2 \log ^3(x-2)}dx+\frac {375}{8} \int \frac {1}{(3 x-5) \log ^3(x-2)}dx-\frac {625}{24} \int \frac {1}{(3 x-5)^3 \log ^2(x-2)}dx-\frac {125}{12} \int \frac {1}{(3 x-5)^2 \log ^2(x-2)}dx-\frac {x (2-x)}{16 \log ^2(x-2)}-\frac {2-x}{3 \log ^2(x-2)}+\frac {9}{\log ^2(x-2)}\) |
Int[(45*x^4 - 27*x^5 + (180*x^3 - 144*x^4 + 27*x^5)*Log[-2 + x])/((2000 - 4600*x + 3960*x^2 - 1512*x^3 + 216*x^4)*Log[-2 + x]^3),x]
3.2.58.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 25.93 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
norman | \(\frac {9 x^{4}}{16 \left (3 x -5\right )^{2} \ln \left (-2+x \right )^{2}}\) | \(19\) |
risch | \(\frac {9 x^{4}}{16 \left (9 x^{2}-30 x +25\right ) \ln \left (-2+x \right )^{2}}\) | \(24\) |
parallelrisch | \(\frac {9 x^{4}}{16 \left (9 x^{2}-30 x +25\right ) \ln \left (-2+x \right )^{2}}\) | \(24\) |
derivativedivides | \(\frac {\frac {125 \left (-2+x \right ) \ln \left (-2+x \right )}{12}+\frac {125 \ln \left (-2+x \right )}{16}-\frac {3125}{72}+\frac {625 x}{24}}{\ln \left (-2+x \right )^{3} \left (3 x -5\right )^{2}}+\frac {19}{16 \ln \left (-2+x \right )^{2}}+\frac {-\frac {11}{12}+\frac {11 x}{24}}{\ln \left (-2+x \right )^{2}}+\frac {\left (-2+x \right )^{2}}{16 \ln \left (-2+x \right )^{2}}-\frac {625}{72 \ln \left (-2+x \right )^{3} \left (3 x -5\right )}\) | \(84\) |
default | \(\frac {\frac {125 \left (-2+x \right ) \ln \left (-2+x \right )}{12}+\frac {125 \ln \left (-2+x \right )}{16}-\frac {3125}{72}+\frac {625 x}{24}}{\ln \left (-2+x \right )^{3} \left (3 x -5\right )^{2}}+\frac {19}{16 \ln \left (-2+x \right )^{2}}+\frac {-\frac {11}{12}+\frac {11 x}{24}}{\ln \left (-2+x \right )^{2}}+\frac {\left (-2+x \right )^{2}}{16 \ln \left (-2+x \right )^{2}}-\frac {625}{72 \ln \left (-2+x \right )^{3} \left (3 x -5\right )}\) | \(84\) |
int(((27*x^5-144*x^4+180*x^3)*ln(-2+x)-27*x^5+45*x^4)/(216*x^4-1512*x^3+39 60*x^2-4600*x+2000)/ln(-2+x)^3,x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \begin {dmath*} \int \frac {45 x^4-27 x^5+\left (180 x^3-144 x^4+27 x^5\right ) \log (-2+x)}{\left (2000-4600 x+3960 x^2-1512 x^3+216 x^4\right ) \log ^3(-2+x)} \, dx=\frac {9 \, x^{4}}{16 \, {\left (9 \, x^{2} - 30 \, x + 25\right )} \log \left (x - 2\right )^{2}} \end {dmath*}
integrate(((27*x^5-144*x^4+180*x^3)*log(-2+x)-27*x^5+45*x^4)/(216*x^4-1512 *x^3+3960*x^2-4600*x+2000)/log(-2+x)^3,x, algorithm=\
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \begin {dmath*} \int \frac {45 x^4-27 x^5+\left (180 x^3-144 x^4+27 x^5\right ) \log (-2+x)}{\left (2000-4600 x+3960 x^2-1512 x^3+216 x^4\right ) \log ^3(-2+x)} \, dx=\frac {9 x^{4}}{\left (144 x^{2} - 480 x + 400\right ) \log {\left (x - 2 \right )}^{2}} \end {dmath*}
integrate(((27*x**5-144*x**4+180*x**3)*ln(-2+x)-27*x**5+45*x**4)/(216*x**4 -1512*x**3+3960*x**2-4600*x+2000)/ln(-2+x)**3,x)
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \begin {dmath*} \int \frac {45 x^4-27 x^5+\left (180 x^3-144 x^4+27 x^5\right ) \log (-2+x)}{\left (2000-4600 x+3960 x^2-1512 x^3+216 x^4\right ) \log ^3(-2+x)} \, dx=\frac {9 \, x^{4}}{16 \, {\left (9 \, x^{2} - 30 \, x + 25\right )} \log \left (x - 2\right )^{2}} \end {dmath*}
integrate(((27*x^5-144*x^4+180*x^3)*log(-2+x)-27*x^5+45*x^4)/(216*x^4-1512 *x^3+3960*x^2-4600*x+2000)/log(-2+x)^3,x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \begin {dmath*} \int \frac {45 x^4-27 x^5+\left (180 x^3-144 x^4+27 x^5\right ) \log (-2+x)}{\left (2000-4600 x+3960 x^2-1512 x^3+216 x^4\right ) \log ^3(-2+x)} \, dx=\frac {9 \, x^{4}}{16 \, {\left (9 \, x^{2} \log \left (x - 2\right )^{2} - 30 \, x \log \left (x - 2\right )^{2} + 25 \, \log \left (x - 2\right )^{2}\right )}} \end {dmath*}
integrate(((27*x^5-144*x^4+180*x^3)*log(-2+x)-27*x^5+45*x^4)/(216*x^4-1512 *x^3+3960*x^2-4600*x+2000)/log(-2+x)^3,x, algorithm=\
Time = 13.25 (sec) , antiderivative size = 158, normalized size of antiderivative = 7.18 \begin {dmath*} \int \frac {45 x^4-27 x^5+\left (180 x^3-144 x^4+27 x^5\right ) \log (-2+x)}{\left (2000-4600 x+3960 x^2-1512 x^3+216 x^4\right ) \log ^3(-2+x)} \, dx=\frac {\frac {9\,x^4}{16\,{\left (3\,x-5\right )}^2}-\frac {9\,x^3\,\ln \left (x-2\right )\,\left (3\,x-10\right )\,\left (x-2\right )}{16\,{\left (3\,x-5\right )}^3}}{{\ln \left (x-2\right )}^2}-\frac {\frac {9\,\left (10\,x^3-3\,x^4\right )\,\left (x-2\right )}{16\,{\left (3\,x-5\right )}^3}-\frac {9\,\ln \left (x-2\right )\,\left (x-2\right )\,\left (-18\,x^5+123\,x^4-320\,x^3+300\,x^2\right )}{16\,{\left (3\,x-5\right )}^4}}{\ln \left (x-2\right )}-\frac {13\,x}{48}+\frac {x^2}{8}+\frac {\frac {125\,x^3}{216}-\frac {1625\,x^2}{648}+\frac {11125\,x}{3888}-\frac {625}{1944}}{x^4-\frac {20\,x^3}{3}+\frac {50\,x^2}{3}-\frac {500\,x}{27}+\frac {625}{81}} \end {dmath*}
int((log(x - 2)*(180*x^3 - 144*x^4 + 27*x^5) + 45*x^4 - 27*x^5)/(log(x - 2 )^3*(3960*x^2 - 4600*x - 1512*x^3 + 216*x^4 + 2000)),x)
((9*x^4)/(16*(3*x - 5)^2) - (9*x^3*log(x - 2)*(3*x - 10)*(x - 2))/(16*(3*x - 5)^3))/log(x - 2)^2 - ((9*(10*x^3 - 3*x^4)*(x - 2))/(16*(3*x - 5)^3) - (9*log(x - 2)*(x - 2)*(300*x^2 - 320*x^3 + 123*x^4 - 18*x^5))/(16*(3*x - 5 )^4))/log(x - 2) - (13*x)/48 + x^2/8 + ((11125*x)/3888 - (1625*x^2)/648 + (125*x^3)/216 - 625/1944)/((50*x^2)/3 - (500*x)/27 - (20*x^3)/3 + x^4 + 62 5/81)