Integrand size = 87, antiderivative size = 28 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\frac {x}{x+x^4 \left (9+\frac {x}{5}+\frac {5}{3 x^4 \log ^2(x)}\right )} \]
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\frac {15 x \log ^2(x)}{25+3 x \left (5+45 x^3+x^4\right ) \log ^2(x)} \]
Integrate[(750*Log[x] + 375*Log[x]^2 + (-6075*x^4 - 180*x^5)*Log[x]^4)/(62 5 + (750*x + 6750*x^4 + 150*x^5)*Log[x]^2 + (225*x^2 + 4050*x^5 + 90*x^6 + 18225*x^8 + 810*x^9 + 9*x^10)*Log[x]^4),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.
| \(\displaystyle \int \frac {\left (-180 x^5-6075 x^4\right ) \log ^4(x)+375 \log ^2(x)+750 \log (x)}{\left (150 x^5+6750 x^4+750 x\right ) \log ^2(x)+\left (9 x^{10}+810 x^9+18225 x^8+90 x^6+4050 x^5+225 x^2\right ) \log ^4(x)+625} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {15 \log (x) \left (-3 (4 x+135) x^4 \log ^3(x)+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 15 \int \left (-\frac {(4 x+135) x^2}{3 \left (x^4+45 x^3+5\right )^2}+\frac {25 \left (9 x^4+315 x^3+5\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right ) x}+\frac {25 \left (6 \log (x) x^9+540 \log (x) x^8+12150 \log (x) x^7+60 \log (x) x^5+2700 \log (x) x^4-125 x^4-4500 x^3+150 \log (x) x-125\right )}{3 \left (x^4+45 x^3+5\right )^2 \left (3 \log ^2(x) x^5+135 \log ^2(x) x^4+15 \log ^2(x) x+25\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 15 \int \frac {\log (x) \left (-3 (4 x+135) \log ^3(x) x^4+25 \log (x)+50\right )}{\left (3 x \left (x^4+45 x^3+5\right ) \log ^2(x)+25\right )^2}dx\) |
Int[(750*Log[x] + 375*Log[x]^2 + (-6075*x^4 - 180*x^5)*Log[x]^4)/(625 + (7 50*x + 6750*x^4 + 150*x^5)*Log[x]^2 + (225*x^2 + 4050*x^5 + 90*x^6 + 18225 *x^8 + 810*x^9 + 9*x^10)*Log[x]^4),x]
3.23.81.3.1 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 = 0.47 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32
| method | result | size |
| default | \(\frac {15 x \ln \left (x \right )^{2}}{3 x^{5} \ln \left (x \right )^{2}+135 x^{4} \ln \left (x \right )^{2}+15 x \ln \left (x \right )^{2}+25}\) | \(37\) |
| parallelrisch | \(\frac {15 x \ln \left (x \right )^{2}}{3 x^{5} \ln \left (x \right )^{2}+135 x^{4} \ln \left (x \right )^{2}+15 x \ln \left (x \right )^{2}+25}\) | \(37\) |
| risch | \(\frac {5}{x^{4}+45 x^{3}+5}-\frac {125}{\left (x^{4}+45 x^{3}+5\right ) \left (3 x^{5} \ln \left (x \right )^{2}+135 x^{4} \ln \left (x \right )^{2}+15 x \ln \left (x \right )^{2}+25\right )}\) | \(59\) |
int(((-180*x^5-6075*x^4)*ln(x)^4+375*ln(x)^2+750*ln(x))/((9*x^10+810*x^9+1 8225*x^8+90*x^6+4050*x^5+225*x^2)*ln(x)^4+(150*x^5+6750*x^4+750*x)*ln(x)^2 +625),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\frac {15 \, x \log \left (x\right )^{2}}{3 \, {\left (x^{5} + 45 \, x^{4} + 5 \, x\right )} \log \left (x\right )^{2} + 25} \]
integrate(((-180*x^5-6075*x^4)*log(x)^4+375*log(x)^2+750*log(x))/((9*x^10+ 810*x^9+18225*x^8+90*x^6+4050*x^5+225*x^2)*log(x)^4+(150*x^5+6750*x^4+750* x)*log(x)^2+625),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=- \frac {125}{25 x^{4} + 1125 x^{3} + \left (3 x^{9} + 270 x^{8} + 6075 x^{7} + 30 x^{5} + 1350 x^{4} + 75 x\right ) \log {\left (x \right )}^{2} + 125} + \frac {5}{x^{4} + 45 x^{3} + 5} \]
integrate(((-180*x**5-6075*x**4)*ln(x)**4+375*ln(x)**2+750*ln(x))/((9*x**1 0+810*x**9+18225*x**8+90*x**6+4050*x**5+225*x**2)*ln(x)**4+(150*x**5+6750* x**4+750*x)*ln(x)**2+625),x)
-125/(25*x**4 + 1125*x**3 + (3*x**9 + 270*x**8 + 6075*x**7 + 30*x**5 + 135 0*x**4 + 75*x)*log(x)**2 + 125) + 5/(x**4 + 45*x**3 + 5)
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\frac {15 \, x \log \left (x\right )^{2}}{3 \, {\left (x^{5} + 45 \, x^{4} + 5 \, x\right )} \log \left (x\right )^{2} + 25} \]
integrate(((-180*x^5-6075*x^4)*log(x)^4+375*log(x)^2+750*log(x))/((9*x^10+ 810*x^9+18225*x^8+90*x^6+4050*x^5+225*x^2)*log(x)^4+(150*x^5+6750*x^4+750* x)*log(x)^2+625),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (27) = 54\).
Time = 0.42 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.96 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=-\frac {125}{3 \, x^{9} \log \left (x\right )^{2} + 270 \, x^{8} \log \left (x\right )^{2} + 6075 \, x^{7} \log \left (x\right )^{2} + 30 \, x^{5} \log \left (x\right )^{2} + 1350 \, x^{4} \log \left (x\right )^{2} + 25 \, x^{4} + 1125 \, x^{3} + 75 \, x \log \left (x\right )^{2} + 125} + \frac {5}{x^{4} + 45 \, x^{3} + 5} \]
integrate(((-180*x^5-6075*x^4)*log(x)^4+375*log(x)^2+750*log(x))/((9*x^10+ 810*x^9+18225*x^8+90*x^6+4050*x^5+225*x^2)*log(x)^4+(150*x^5+6750*x^4+750* x)*log(x)^2+625),x, algorithm=\
-125/(3*x^9*log(x)^2 + 270*x^8*log(x)^2 + 6075*x^7*log(x)^2 + 30*x^5*log(x )^2 + 1350*x^4*log(x)^2 + 25*x^4 + 1125*x^3 + 75*x*log(x)^2 + 125) + 5/(x^ 4 + 45*x^3 + 5)
Timed out. \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\int \frac {\left (-180\,x^5-6075\,x^4\right )\,{\ln \left (x\right )}^4+375\,{\ln \left (x\right )}^2+750\,\ln \left (x\right )}{\left (9\,x^{10}+810\,x^9+18225\,x^8+90\,x^6+4050\,x^5+225\,x^2\right )\,{\ln \left (x\right )}^4+\left (150\,x^5+6750\,x^4+750\,x\right )\,{\ln \left (x\right )}^2+625} \,d x \]
int((750*log(x) + 375*log(x)^2 - log(x)^4*(6075*x^4 + 180*x^5))/(log(x)^2* (750*x + 6750*x^4 + 150*x^5) + log(x)^4*(225*x^2 + 4050*x^5 + 90*x^6 + 182 25*x^8 + 810*x^9 + 9*x^10) + 625),x)