Integrand size = 122, antiderivative size = 23 \[ \int \frac {-14950+17450 x+32400 x^2+209952 x^3+\left (7475 x+88776 x^2-104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )}{14950 x+177552 x^2-209952 x^3+\left (-7475 x-88776 x^2+104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )} \, dx=-x+\log \left (-2+\log \left (\left (-x+\frac {x}{\frac {25}{324}+x}\right )^2\right )\right ) \]
Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {-14950+17450 x+32400 x^2+209952 x^3+\left (7475 x+88776 x^2-104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )}{14950 x+177552 x^2-209952 x^3+\left (-7475 x-88776 x^2+104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )} \, dx=-x+\log \left (2-\log \left (\frac {x^2 (-299+324 x)^2}{(25+324 x)^2}\right )\right ) \]
Integrate[(-14950 + 17450*x + 32400*x^2 + 209952*x^3 + (7475*x + 88776*x^2 - 104976*x^3)*Log[(89401*x^2 - 193752*x^3 + 104976*x^4)/(625 + 16200*x + 104976*x^2)])/(14950*x + 177552*x^2 - 209952*x^3 + (-7475*x - 88776*x^2 + 104976*x^3)*Log[(89401*x^2 - 193752*x^3 + 104976*x^4)/(625 + 16200*x + 104 976*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.
| \(\displaystyle \int \frac {209952 x^3+32400 x^2+\left (-104976 x^3+88776 x^2+7475 x\right ) \log \left (\frac {104976 x^4-193752 x^3+89401 x^2}{104976 x^2+16200 x+625}\right )+17450 x-14950}{-209952 x^3+177552 x^2+\left (104976 x^3-88776 x^2-7475 x\right ) \log \left (\frac {104976 x^4-193752 x^3+89401 x^2}{104976 x^2+16200 x+625}\right )+14950 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {209952 x^3+32400 x^2+\left (-104976 x^3+88776 x^2+7475 x\right ) \log \left (\frac {104976 x^4-193752 x^3+89401 x^2}{104976 x^2+16200 x+625}\right )+17450 x-14950}{x \left (-104976 x^2+88776 x+7475\right ) \left (2-\log \left (\frac {x^2 (324 x-299)^2}{(324 x+25)^2}\right )\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {209952 x^2}{(324 x-299) (324 x+25) \left (\log \left (\frac {x^2 (324 x-299)^2}{(324 x+25)^2}\right )-2\right )}+\frac {32400 x}{(324 x-299) (324 x+25) \left (\log \left (\frac {x^2 (324 x-299)^2}{(324 x+25)^2}\right )-2\right )}-\frac {\log \left (\frac {x^2 (324 x-299)^2}{(324 x+25)^2}\right )}{\log \left (\frac {x^2 (324 x-299)^2}{(324 x+25)^2}\right )-2}+\frac {17450}{(324 x-299) (324 x+25) \left (\log \left (\frac {x^2 (324 x-299)^2}{(324 x+25)^2}\right )-2\right )}-\frac {14950}{(324 x-299) (324 x+25) x \left (\log \left (\frac {x^2 (324 x-299)^2}{(324 x+25)^2}\right )-2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {1}{x \left (\log \left (\frac {x^2 (324 x-299)^2}{(324 x+25)^2}\right )-2\right )}dx+648 \int \frac {1}{(324 x-299) \left (\log \left (\frac {x^2 (324 x-299)^2}{(324 x+25)^2}\right )-2\right )}dx-648 \int \frac {1}{(324 x+25) \left (\log \left (\frac {x^2 (324 x-299)^2}{(324 x+25)^2}\right )-2\right )}dx-x\) |
Int[(-14950 + 17450*x + 32400*x^2 + 209952*x^3 + (7475*x + 88776*x^2 - 104 976*x^3)*Log[(89401*x^2 - 193752*x^3 + 104976*x^4)/(625 + 16200*x + 104976 *x^2)])/(14950*x + 177552*x^2 - 209952*x^3 + (-7475*x - 88776*x^2 + 104976 *x^3)*Log[(89401*x^2 - 193752*x^3 + 104976*x^4)/(625 + 16200*x + 104976*x^ 2)]),x]
3.13.49.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.52 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57
| method | result | size |
| parallelrisch | \(-\frac {137}{81}+\ln \left (\ln \left (\frac {x^{2} \left (104976 x^{2}-193752 x +89401\right )}{104976 x^{2}+16200 x +625}\right )-2\right )-x\) | \(36\) |
| norman | \(-x +\ln \left (\ln \left (\frac {104976 x^{4}-193752 x^{3}+89401 x^{2}}{104976 x^{2}+16200 x +625}\right )-2\right )\) | \(38\) |
| risch | \(-x +\ln \left (\ln \left (\frac {104976 x^{4}-193752 x^{3}+89401 x^{2}}{104976 x^{2}+16200 x +625}\right )-2\right )\) | \(38\) |
int(((-104976*x^3+88776*x^2+7475*x)*ln((104976*x^4-193752*x^3+89401*x^2)/( 104976*x^2+16200*x+625))+209952*x^3+32400*x^2+17450*x-14950)/((104976*x^3- 88776*x^2-7475*x)*ln((104976*x^4-193752*x^3+89401*x^2)/(104976*x^2+16200*x +625))-209952*x^3+177552*x^2+14950*x),x,method=_RETURNVERBOSE)
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {-14950+17450 x+32400 x^2+209952 x^3+\left (7475 x+88776 x^2-104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )}{14950 x+177552 x^2-209952 x^3+\left (-7475 x-88776 x^2+104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )} \, dx=-x + \log \left (\log \left (\frac {104976 \, x^{4} - 193752 \, x^{3} + 89401 \, x^{2}}{104976 \, x^{2} + 16200 \, x + 625}\right ) - 2\right ) \]
integrate(((-104976*x^3+88776*x^2+7475*x)*log((104976*x^4-193752*x^3+89401 *x^2)/(104976*x^2+16200*x+625))+209952*x^3+32400*x^2+17450*x-14950)/((1049 76*x^3-88776*x^2-7475*x)*log((104976*x^4-193752*x^3+89401*x^2)/(104976*x^2 +16200*x+625))-209952*x^3+177552*x^2+14950*x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {-14950+17450 x+32400 x^2+209952 x^3+\left (7475 x+88776 x^2-104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )}{14950 x+177552 x^2-209952 x^3+\left (-7475 x-88776 x^2+104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )} \, dx=- x + \log {\left (\log {\left (\frac {104976 x^{4} - 193752 x^{3} + 89401 x^{2}}{104976 x^{2} + 16200 x + 625} \right )} - 2 \right )} \]
integrate(((-104976*x**3+88776*x**2+7475*x)*ln((104976*x**4-193752*x**3+89 401*x**2)/(104976*x**2+16200*x+625))+209952*x**3+32400*x**2+17450*x-14950) /((104976*x**3-88776*x**2-7475*x)*ln((104976*x**4-193752*x**3+89401*x**2)/ (104976*x**2+16200*x+625))-209952*x**3+177552*x**2+14950*x),x)
Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-14950+17450 x+32400 x^2+209952 x^3+\left (7475 x+88776 x^2-104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )}{14950 x+177552 x^2-209952 x^3+\left (-7475 x-88776 x^2+104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )} \, dx=-x + \log \left (\log \left (324 \, x + 25\right ) - \log \left (324 \, x - 299\right ) - \log \left (x\right ) + 1\right ) \]
integrate(((-104976*x^3+88776*x^2+7475*x)*log((104976*x^4-193752*x^3+89401 *x^2)/(104976*x^2+16200*x+625))+209952*x^3+32400*x^2+17450*x-14950)/((1049 76*x^3-88776*x^2-7475*x)*log((104976*x^4-193752*x^3+89401*x^2)/(104976*x^2 +16200*x+625))-209952*x^3+177552*x^2+14950*x),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {-14950+17450 x+32400 x^2+209952 x^3+\left (7475 x+88776 x^2-104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )}{14950 x+177552 x^2-209952 x^3+\left (-7475 x-88776 x^2+104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )} \, dx=-x + \log \left (\log \left (\frac {104976 \, x^{4} - 193752 \, x^{3} + 89401 \, x^{2}}{104976 \, x^{2} + 16200 \, x + 625}\right ) - 2\right ) \]
integrate(((-104976*x^3+88776*x^2+7475*x)*log((104976*x^4-193752*x^3+89401 *x^2)/(104976*x^2+16200*x+625))+209952*x^3+32400*x^2+17450*x-14950)/((1049 76*x^3-88776*x^2-7475*x)*log((104976*x^4-193752*x^3+89401*x^2)/(104976*x^2 +16200*x+625))-209952*x^3+177552*x^2+14950*x),x, algorithm=\
Time = 8.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {-14950+17450 x+32400 x^2+209952 x^3+\left (7475 x+88776 x^2-104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )}{14950 x+177552 x^2-209952 x^3+\left (-7475 x-88776 x^2+104976 x^3\right ) \log \left (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2}\right )} \, dx=\ln \left (\ln \left (\frac {104976\,x^4-193752\,x^3+89401\,x^2}{104976\,x^2+16200\,x+625}\right )-2\right )-x \]
int((17450*x + log((89401*x^2 - 193752*x^3 + 104976*x^4)/(16200*x + 104976 *x^2 + 625))*(7475*x + 88776*x^2 - 104976*x^3) + 32400*x^2 + 209952*x^3 - 14950)/(14950*x - log((89401*x^2 - 193752*x^3 + 104976*x^4)/(16200*x + 104 976*x^2 + 625))*(7475*x + 88776*x^2 - 104976*x^3) + 177552*x^2 - 209952*x^ 3),x)