Integrand size = 128, antiderivative size = 30 \[ \int \frac {21600+76375 x-23350 x^2+\left (-15000-61000 x+19400 x^2\right ) \log (x)+\left (3600+18500 x-6200 x^2\right ) \log ^2(x)+\left (-300-2500 x+900 x^2\right ) \log ^3(x)+\left (125 x-50 x^2\right ) \log ^4(x)}{312987 x-558144 x \log (x)+442632 x \log ^2(x)-203808 x \log ^3(x)+59586 x \log ^4(x)-11328 x \log ^5(x)+1368 x \log ^6(x)-96 x \log ^7(x)+3 x \log ^8(x)} \, dx=\frac {25 \left (1+\frac {1}{3} (5-x) x\right )}{-1+\left (2+(4-\log (x))^2\right )^2} \]
Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {21600+76375 x-23350 x^2+\left (-15000-61000 x+19400 x^2\right ) \log (x)+\left (3600+18500 x-6200 x^2\right ) \log ^2(x)+\left (-300-2500 x+900 x^2\right ) \log ^3(x)+\left (125 x-50 x^2\right ) \log ^4(x)}{312987 x-558144 x \log (x)+442632 x \log ^2(x)-203808 x \log ^3(x)+59586 x \log ^4(x)-11328 x \log ^5(x)+1368 x \log ^6(x)-96 x \log ^7(x)+3 x \log ^8(x)} \, dx=-\frac {25 \left (-3-5 x+x^2\right )}{3 \left (323-288 \log (x)+100 \log ^2(x)-16 \log ^3(x)+\log ^4(x)\right )} \]
Integrate[(21600 + 76375*x - 23350*x^2 + (-15000 - 61000*x + 19400*x^2)*Lo g[x] + (3600 + 18500*x - 6200*x^2)*Log[x]^2 + (-300 - 2500*x + 900*x^2)*Lo g[x]^3 + (125*x - 50*x^2)*Log[x]^4)/(312987*x - 558144*x*Log[x] + 442632*x *Log[x]^2 - 203808*x*Log[x]^3 + 59586*x*Log[x]^4 - 11328*x*Log[x]^5 + 1368 *x*Log[x]^6 - 96*x*Log[x]^7 + 3*x*Log[x]^8),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.
| \(\displaystyle \int \frac {-23350 x^2+\left (125 x-50 x^2\right ) \log ^4(x)+\left (900 x^2-2500 x-300\right ) \log ^3(x)+\left (-6200 x^2+18500 x+3600\right ) \log ^2(x)+\left (19400 x^2-61000 x-15000\right ) \log (x)+76375 x+21600}{312987 x+3 x \log ^8(x)-96 x \log ^7(x)+1368 x \log ^6(x)-11328 x \log ^5(x)+59586 x \log ^4(x)-203808 x \log ^3(x)+442632 x \log ^2(x)-558144 x \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-23350 x^2+\left (125 x-50 x^2\right ) \log ^4(x)+\left (900 x^2-2500 x-300\right ) \log ^3(x)+\left (-6200 x^2+18500 x+3600\right ) \log ^2(x)+\left (19400 x^2-61000 x-15000\right ) \log (x)+76375 x+21600}{3 x \left (\log ^4(x)-16 \log ^3(x)+100 \log ^2(x)-288 \log (x)+323\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {25 \left (\left (5 x-2 x^2\right ) \log ^4(x)-4 \left (-9 x^2+25 x+3\right ) \log ^3(x)+4 \left (-62 x^2+185 x+36\right ) \log ^2(x)-8 \left (-97 x^2+305 x+75\right ) \log (x)-934 x^2+3055 x+864\right )}{x \left (\log ^4(x)-16 \log ^3(x)+100 \log ^2(x)-288 \log (x)+323\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {25}{3} \int \frac {\left (5 x-2 x^2\right ) \log ^4(x)-4 \left (-9 x^2+25 x+3\right ) \log ^3(x)+4 \left (-62 x^2+185 x+36\right ) \log ^2(x)-8 \left (-97 x^2+305 x+75\right ) \log (x)-934 x^2+3055 x+864}{x \left (\log ^4(x)-16 \log ^3(x)+100 \log ^2(x)-288 \log (x)+323\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {25}{3} \int \left (\frac {5-2 x}{2 \left (\log ^2(x)-8 \log (x)+17\right )}+\frac {2 x-5}{2 \left (\log ^2(x)-8 \log (x)+19\right )}+\frac {\left (x^2-5 x-3\right ) (\log (x)-4)}{x \left (\log ^2(x)-8 \log (x)+17\right )^2}-\frac {\left (x^2-5 x-3\right ) (\log (x)-4)}{x \left (\log ^2(x)-8 \log (x)+19\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {25}{3} \left (-4 \int \frac {x}{\left (\log ^2(x)-8 \log (x)+17\right )^2}dx+\int \frac {x \log (x)}{\left (\log ^2(x)-8 \log (x)+17\right )^2}dx-\int \frac {x}{\log ^2(x)-8 \log (x)+17}dx+4 \int \frac {x}{\left (\log ^2(x)-8 \log (x)+19\right )^2}dx-\int \frac {x \log (x)}{\left (\log ^2(x)-8 \log (x)+19\right )^2}dx+\int \frac {x}{\log ^2(x)-8 \log (x)+19}dx-\frac {5}{12} \left (4+i \sqrt {3}\right ) e^{4+i \sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)-i \sqrt {3}-4\right )+\frac {5 i e^{4+i \sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)-i \sqrt {3}-4\right )}{4 \sqrt {3}}+\frac {5}{3} e^{4+i \sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)-i \sqrt {3}-4\right )-\frac {5}{12} \left (4-i \sqrt {3}\right ) e^{4-i \sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)+i \sqrt {3}-4\right )-\frac {5 i e^{4-i \sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)+i \sqrt {3}-4\right )}{4 \sqrt {3}}+\frac {5}{3} e^{4-i \sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)+i \sqrt {3}-4\right )+\frac {3}{2 \left (\log ^2(x)-8 \log (x)+17\right )}-\frac {3}{2 \left (\log ^2(x)-8 \log (x)+19\right )}+\frac {5 i x}{2 (-2 \log (x)+(8+2 i))}-\frac {5 \left (4-i \sqrt {3}\right ) x}{12 \left (-\log (x)-i \sqrt {3}+4\right )}+\frac {5 x}{3 \left (-\log (x)-i \sqrt {3}+4\right )}-\frac {5 \left (4+i \sqrt {3}\right ) x}{12 \left (-\log (x)+i \sqrt {3}+4\right )}+\frac {5 x}{3 \left (-\log (x)+i \sqrt {3}+4\right )}+\frac {5 i x}{2 (2 \log (x)-(8-2 i))}\right )\) |
Int[(21600 + 76375*x - 23350*x^2 + (-15000 - 61000*x + 19400*x^2)*Log[x] + (3600 + 18500*x - 6200*x^2)*Log[x]^2 + (-300 - 2500*x + 900*x^2)*Log[x]^3 + (125*x - 50*x^2)*Log[x]^4)/(312987*x - 558144*x*Log[x] + 442632*x*Log[x ]^2 - 203808*x*Log[x]^3 + 59586*x*Log[x]^4 - 11328*x*Log[x]^5 + 1368*x*Log [x]^6 - 96*x*Log[x]^7 + 3*x*Log[x]^8),x]
3.10.31.3.1 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 = 0.97 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17
| method | result | size |
| risch | \(-\frac {25 \left (x^{2}-5 x -3\right )}{3 \left (\ln \left (x \right )^{4}-16 \ln \left (x \right )^{3}+100 \ln \left (x \right )^{2}-288 \ln \left (x \right )+323\right )}\) | \(35\) |
| parallelrisch | \(\frac {-25 x^{2}+125 x +75}{3 \ln \left (x \right )^{4}-48 \ln \left (x \right )^{3}+300 \ln \left (x \right )^{2}-864 \ln \left (x \right )+969}\) | \(37\) |
| default | \(\text {Expression too large to display}\) | \(622\) |
int(((-50*x^2+125*x)*ln(x)^4+(900*x^2-2500*x-300)*ln(x)^3+(-6200*x^2+18500 *x+3600)*ln(x)^2+(19400*x^2-61000*x-15000)*ln(x)-23350*x^2+76375*x+21600)/ (3*x*ln(x)^8-96*x*ln(x)^7+1368*x*ln(x)^6-11328*x*ln(x)^5+59586*x*ln(x)^4-2 03808*x*ln(x)^3+442632*x*ln(x)^2-558144*x*ln(x)+312987*x),x,method=_RETURN VERBOSE)
Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {21600+76375 x-23350 x^2+\left (-15000-61000 x+19400 x^2\right ) \log (x)+\left (3600+18500 x-6200 x^2\right ) \log ^2(x)+\left (-300-2500 x+900 x^2\right ) \log ^3(x)+\left (125 x-50 x^2\right ) \log ^4(x)}{312987 x-558144 x \log (x)+442632 x \log ^2(x)-203808 x \log ^3(x)+59586 x \log ^4(x)-11328 x \log ^5(x)+1368 x \log ^6(x)-96 x \log ^7(x)+3 x \log ^8(x)} \, dx=-\frac {25 \, {\left (x^{2} - 5 \, x - 3\right )}}{3 \, {\left (\log \left (x\right )^{4} - 16 \, \log \left (x\right )^{3} + 100 \, \log \left (x\right )^{2} - 288 \, \log \left (x\right ) + 323\right )}} \]
integrate(((-50*x^2+125*x)*log(x)^4+(900*x^2-2500*x-300)*log(x)^3+(-6200*x ^2+18500*x+3600)*log(x)^2+(19400*x^2-61000*x-15000)*log(x)-23350*x^2+76375 *x+21600)/(3*x*log(x)^8-96*x*log(x)^7+1368*x*log(x)^6-11328*x*log(x)^5+595 86*x*log(x)^4-203808*x*log(x)^3+442632*x*log(x)^2-558144*x*log(x)+312987*x ),x, algorithm=\
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {21600+76375 x-23350 x^2+\left (-15000-61000 x+19400 x^2\right ) \log (x)+\left (3600+18500 x-6200 x^2\right ) \log ^2(x)+\left (-300-2500 x+900 x^2\right ) \log ^3(x)+\left (125 x-50 x^2\right ) \log ^4(x)}{312987 x-558144 x \log (x)+442632 x \log ^2(x)-203808 x \log ^3(x)+59586 x \log ^4(x)-11328 x \log ^5(x)+1368 x \log ^6(x)-96 x \log ^7(x)+3 x \log ^8(x)} \, dx=\frac {- 25 x^{2} + 125 x + 75}{3 \log {\left (x \right )}^{4} - 48 \log {\left (x \right )}^{3} + 300 \log {\left (x \right )}^{2} - 864 \log {\left (x \right )} + 969} \]
integrate(((-50*x**2+125*x)*ln(x)**4+(900*x**2-2500*x-300)*ln(x)**3+(-6200 *x**2+18500*x+3600)*ln(x)**2+(19400*x**2-61000*x-15000)*ln(x)-23350*x**2+7 6375*x+21600)/(3*x*ln(x)**8-96*x*ln(x)**7+1368*x*ln(x)**6-11328*x*ln(x)**5 +59586*x*ln(x)**4-203808*x*ln(x)**3+442632*x*ln(x)**2-558144*x*ln(x)+31298 7*x),x)
Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {21600+76375 x-23350 x^2+\left (-15000-61000 x+19400 x^2\right ) \log (x)+\left (3600+18500 x-6200 x^2\right ) \log ^2(x)+\left (-300-2500 x+900 x^2\right ) \log ^3(x)+\left (125 x-50 x^2\right ) \log ^4(x)}{312987 x-558144 x \log (x)+442632 x \log ^2(x)-203808 x \log ^3(x)+59586 x \log ^4(x)-11328 x \log ^5(x)+1368 x \log ^6(x)-96 x \log ^7(x)+3 x \log ^8(x)} \, dx=-\frac {25 \, {\left (x^{2} - 5 \, x - 3\right )}}{3 \, {\left (\log \left (x\right )^{4} - 16 \, \log \left (x\right )^{3} + 100 \, \log \left (x\right )^{2} - 288 \, \log \left (x\right ) + 323\right )}} \]
integrate(((-50*x^2+125*x)*log(x)^4+(900*x^2-2500*x-300)*log(x)^3+(-6200*x ^2+18500*x+3600)*log(x)^2+(19400*x^2-61000*x-15000)*log(x)-23350*x^2+76375 *x+21600)/(3*x*log(x)^8-96*x*log(x)^7+1368*x*log(x)^6-11328*x*log(x)^5+595 86*x*log(x)^4-203808*x*log(x)^3+442632*x*log(x)^2-558144*x*log(x)+312987*x ),x, algorithm=\
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {21600+76375 x-23350 x^2+\left (-15000-61000 x+19400 x^2\right ) \log (x)+\left (3600+18500 x-6200 x^2\right ) \log ^2(x)+\left (-300-2500 x+900 x^2\right ) \log ^3(x)+\left (125 x-50 x^2\right ) \log ^4(x)}{312987 x-558144 x \log (x)+442632 x \log ^2(x)-203808 x \log ^3(x)+59586 x \log ^4(x)-11328 x \log ^5(x)+1368 x \log ^6(x)-96 x \log ^7(x)+3 x \log ^8(x)} \, dx=-\frac {25 \, {\left (x^{2} - 5 \, x - 3\right )}}{3 \, {\left (\log \left (x\right )^{4} - 16 \, \log \left (x\right )^{3} + 100 \, \log \left (x\right )^{2} - 288 \, \log \left (x\right ) + 323\right )}} \]
integrate(((-50*x^2+125*x)*log(x)^4+(900*x^2-2500*x-300)*log(x)^3+(-6200*x ^2+18500*x+3600)*log(x)^2+(19400*x^2-61000*x-15000)*log(x)-23350*x^2+76375 *x+21600)/(3*x*log(x)^8-96*x*log(x)^7+1368*x*log(x)^6-11328*x*log(x)^5+595 86*x*log(x)^4-203808*x*log(x)^3+442632*x*log(x)^2-558144*x*log(x)+312987*x ),x, algorithm=\
Time = 13.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {21600+76375 x-23350 x^2+\left (-15000-61000 x+19400 x^2\right ) \log (x)+\left (3600+18500 x-6200 x^2\right ) \log ^2(x)+\left (-300-2500 x+900 x^2\right ) \log ^3(x)+\left (125 x-50 x^2\right ) \log ^4(x)}{312987 x-558144 x \log (x)+442632 x \log ^2(x)-203808 x \log ^3(x)+59586 x \log ^4(x)-11328 x \log ^5(x)+1368 x \log ^6(x)-96 x \log ^7(x)+3 x \log ^8(x)} \, dx=\frac {-\frac {25\,x^2}{6}+\frac {125\,x}{6}+\frac {25}{2}}{{\ln \left (x\right )}^2-8\,\ln \left (x\right )+17}-\frac {-\frac {25\,x^2}{6}+\frac {125\,x}{6}+\frac {25}{2}}{{\ln \left (x\right )}^2-8\,\ln \left (x\right )+19} \]
int((76375*x + log(x)^4*(125*x - 50*x^2) - log(x)^3*(2500*x - 900*x^2 + 30 0) + log(x)^2*(18500*x - 6200*x^2 + 3600) - log(x)*(61000*x - 19400*x^2 + 15000) - 23350*x^2 + 21600)/(312987*x + 442632*x*log(x)^2 - 203808*x*log(x )^3 + 59586*x*log(x)^4 - 11328*x*log(x)^5 + 1368*x*log(x)^6 - 96*x*log(x)^ 7 + 3*x*log(x)^8 - 558144*x*log(x)),x)