Integrand size = 105, antiderivative size = 26 \[ \int \frac {-375 x+15 x^2+\left (10125+10845 x+4925 x^2+1035 x^3+75 x^4-5 x^5\right ) \log (-25+x)+\left (-405 x-450 x^2-215 x^3-50 x^4-5 x^5+\left (11250 x+10300 x^2+3320 x^3+350 x^4-20 x^5\right ) \log (-25+x)\right ) \log (x)}{-25 x+x^2} \, dx=5 \left (-6+3 x-\left (x-(3+x)^2\right )^2 \log (-25+x) \log (x)\right ) \]
Time = 0.71 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-375 x+15 x^2+\left (10125+10845 x+4925 x^2+1035 x^3+75 x^4-5 x^5\right ) \log (-25+x)+\left (-405 x-450 x^2-215 x^3-50 x^4-5 x^5+\left (11250 x+10300 x^2+3320 x^3+350 x^4-20 x^5\right ) \log (-25+x)\right ) \log (x)}{-25 x+x^2} \, dx=-5 \left (-3 x+\left (9+5 x+x^2\right )^2 \log (-25+x) \log (x)\right ) \]
Integrate[(-375*x + 15*x^2 + (10125 + 10845*x + 4925*x^2 + 1035*x^3 + 75*x ^4 - 5*x^5)*Log[-25 + x] + (-405*x - 450*x^2 - 215*x^3 - 50*x^4 - 5*x^5 + (11250*x + 10300*x^2 + 3320*x^3 + 350*x^4 - 20*x^5)*Log[-25 + x])*Log[x])/ (-25*x + x^2),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.37 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2026, 7293, 2009}
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.
\(\displaystyle \int \frac {15 x^2+\left (-5 x^5+75 x^4+1035 x^3+4925 x^2+10845 x+10125\right ) \log (x-25)+\left (-5 x^5-50 x^4-215 x^3-450 x^2+\left (-20 x^5+350 x^4+3320 x^3+10300 x^2+11250 x\right ) \log (x-25)-405 x\right ) \log (x)-375 x}{x^2-25 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {15 x^2+\left (-5 x^5+75 x^4+1035 x^3+4925 x^2+10845 x+10125\right ) \log (x-25)+\left (-5 x^5-50 x^4-215 x^3-450 x^2+\left (-20 x^5+350 x^4+3320 x^3+10300 x^2+11250 x\right ) \log (x-25)-405 x\right ) \log (x)-375 x}{(x-25) x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {5 \left (x^2+5 x+9\right ) \left (x^2+4 x^2 \log (x-25)+5 x-90 x \log (x-25)-250 \log (x-25)+9\right ) \log (x)}{x-25}-\frac {5 \left (x^4 \log (x-25)+10 x^3 \log (x-25)+43 x^2 \log (x-25)-3 x+90 x \log (x-25)+81 \log (x-25)\right )}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2868750 \operatorname {PolyLog}\left (2,1-\frac {x}{25}\right )+2868750 \operatorname {PolyLog}\left (2,\frac {x}{25}\right )-5 x^4 \log (x-25) \log (x)-50 x^3 \log (x-25) \log (x)-215 x^2 \log (x-25) \log (x)+15 x-2880405 \log (25) \log (x-25)-11655 \log (x-25) \log \left (\frac {x}{25}\right )+2868750 \log (25-x) \log (x)+450 (25-x) \log (x-25) \log (x)-2868750 \log (25) \log (x)\) |
Int[(-375*x + 15*x^2 + (10125 + 10845*x + 4925*x^2 + 1035*x^3 + 75*x^4 - 5 *x^5)*Log[-25 + x] + (-405*x - 450*x^2 - 215*x^3 - 50*x^4 - 5*x^5 + (11250 *x + 10300*x^2 + 3320*x^3 + 350*x^4 - 20*x^5)*Log[-25 + x])*Log[x])/(-25*x + x^2),x]
15*x - 2880405*Log[25]*Log[-25 + x] - 11655*Log[-25 + x]*Log[x/25] - 28687 50*Log[25]*Log[x] + 2868750*Log[25 - x]*Log[x] + 450*(25 - x)*Log[-25 + x] *Log[x] - 215*x^2*Log[-25 + x]*Log[x] - 50*x^3*Log[-25 + x]*Log[x] - 5*x^4 *Log[-25 + x]*Log[x] + 2868750*PolyLog[2, 1 - x/25] + 2868750*PolyLog[2, x /25]
3.18.76.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
risch | \(-5 \left (x^{2}+5 x +9\right )^{2} \ln \left (x \right ) \ln \left (x -25\right )+15 x\) | \(23\) |
parallelrisch | \(-5 \ln \left (x \right ) \ln \left (x -25\right ) x^{4}-50 \ln \left (x \right ) \ln \left (x -25\right ) x^{3}-215 \ln \left (x \right ) \ln \left (x -25\right ) x^{2}-450 \ln \left (x \right ) \ln \left (x -25\right ) x -405 \ln \left (x \right ) \ln \left (x -25\right )+15 x +\frac {375}{2}\) | \(56\) |
default | \(15 x -5 \left (\left (-\frac {1}{4}+\ln \left (x \right )\right ) x^{4}+\left (-\frac {10}{3}+10 \ln \left (x \right )\right ) x^{3}+\left (-\frac {43}{2}+43 \ln \left (x \right )\right ) x^{2}+\left (-90+90 \ln \left (x \right )\right ) x \right ) \ln \left (x -25\right )-405 \left (\ln \left (x \right )-\ln \left (\frac {x}{25}\right )\right ) \ln \left (-\frac {x}{25}+1\right )-\frac {9925625 \ln \left (x -25\right )}{12}-\frac {5 \left (x -25\right )^{4} \ln \left (x -25\right )}{4}-\frac {231366875}{144}-\frac {425 \left (x -25\right )^{3} \ln \left (x -25\right )}{3}-6045 \left (x -25\right )^{2} \ln \left (x -25\right )-115200 \left (x -25\right ) \ln \left (x -25\right )-405 \ln \left (x -25\right ) \ln \left (\frac {x}{25}\right )\) | \(124\) |
parts | \(15 x -5 \left (\left (-\frac {1}{4}+\ln \left (x \right )\right ) x^{4}+\left (-\frac {10}{3}+10 \ln \left (x \right )\right ) x^{3}+\left (-\frac {43}{2}+43 \ln \left (x \right )\right ) x^{2}+\left (-90+90 \ln \left (x \right )\right ) x \right ) \ln \left (x -25\right )-405 \left (\ln \left (x \right )-\ln \left (\frac {x}{25}\right )\right ) \ln \left (-\frac {x}{25}+1\right )-\frac {9925625 \ln \left (x -25\right )}{12}-\frac {5 \left (x -25\right )^{4} \ln \left (x -25\right )}{4}-\frac {231366875}{144}-\frac {425 \left (x -25\right )^{3} \ln \left (x -25\right )}{3}-6045 \left (x -25\right )^{2} \ln \left (x -25\right )-115200 \left (x -25\right ) \ln \left (x -25\right )-405 \ln \left (x -25\right ) \ln \left (\frac {x}{25}\right )\) | \(124\) |
int((((-20*x^5+350*x^4+3320*x^3+10300*x^2+11250*x)*ln(x-25)-5*x^5-50*x^4-2 15*x^3-450*x^2-405*x)*ln(x)+(-5*x^5+75*x^4+1035*x^3+4925*x^2+10845*x+10125 )*ln(x-25)+15*x^2-375*x)/(x^2-25*x),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-375 x+15 x^2+\left (10125+10845 x+4925 x^2+1035 x^3+75 x^4-5 x^5\right ) \log (-25+x)+\left (-405 x-450 x^2-215 x^3-50 x^4-5 x^5+\left (11250 x+10300 x^2+3320 x^3+350 x^4-20 x^5\right ) \log (-25+x)\right ) \log (x)}{-25 x+x^2} \, dx=-5 \, {\left (x^{4} + 10 \, x^{3} + 43 \, x^{2} + 90 \, x + 81\right )} \log \left (x - 25\right ) \log \left (x\right ) + 15 \, x \]
integrate((((-20*x^5+350*x^4+3320*x^3+10300*x^2+11250*x)*log(x-25)-5*x^5-5 0*x^4-215*x^3-450*x^2-405*x)*log(x)+(-5*x^5+75*x^4+1035*x^3+4925*x^2+10845 *x+10125)*log(x-25)+15*x^2-375*x)/(x^2-25*x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {-375 x+15 x^2+\left (10125+10845 x+4925 x^2+1035 x^3+75 x^4-5 x^5\right ) \log (-25+x)+\left (-405 x-450 x^2-215 x^3-50 x^4-5 x^5+\left (11250 x+10300 x^2+3320 x^3+350 x^4-20 x^5\right ) \log (-25+x)\right ) \log (x)}{-25 x+x^2} \, dx=15 x + \left (- 5 x^{4} \log {\left (x \right )} - 50 x^{3} \log {\left (x \right )} - 215 x^{2} \log {\left (x \right )} - 450 x \log {\left (x \right )} - 405 \log {\left (x \right )}\right ) \log {\left (x - 25 \right )} \]
integrate((((-20*x**5+350*x**4+3320*x**3+10300*x**2+11250*x)*ln(x-25)-5*x* *5-50*x**4-215*x**3-450*x**2-405*x)*ln(x)+(-5*x**5+75*x**4+1035*x**3+4925* x**2+10845*x+10125)*ln(x-25)+15*x**2-375*x)/(x**2-25*x),x)
15*x + (-5*x**4*log(x) - 50*x**3*log(x) - 215*x**2*log(x) - 450*x*log(x) - 405*log(x))*log(x - 25)
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-375 x+15 x^2+\left (10125+10845 x+4925 x^2+1035 x^3+75 x^4-5 x^5\right ) \log (-25+x)+\left (-405 x-450 x^2-215 x^3-50 x^4-5 x^5+\left (11250 x+10300 x^2+3320 x^3+350 x^4-20 x^5\right ) \log (-25+x)\right ) \log (x)}{-25 x+x^2} \, dx=-5 \, {\left (x^{4} + 10 \, x^{3} + 43 \, x^{2} + 90 \, x + 81\right )} \log \left (x - 25\right ) \log \left (x\right ) + 15 \, x \]
integrate((((-20*x^5+350*x^4+3320*x^3+10300*x^2+11250*x)*log(x-25)-5*x^5-5 0*x^4-215*x^3-450*x^2-405*x)*log(x)+(-5*x^5+75*x^4+1035*x^3+4925*x^2+10845 *x+10125)*log(x-25)+15*x^2-375*x)/(x^2-25*x),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {-375 x+15 x^2+\left (10125+10845 x+4925 x^2+1035 x^3+75 x^4-5 x^5\right ) \log (-25+x)+\left (-405 x-450 x^2-215 x^3-50 x^4-5 x^5+\left (11250 x+10300 x^2+3320 x^3+350 x^4-20 x^5\right ) \log (-25+x)\right ) \log (x)}{-25 x+x^2} \, dx=-5 \, {\left ({\left (x^{4} + 10 \, x^{3} + 43 \, x^{2} + 90 \, x\right )} \log \left (x\right ) + 81 \, \log \left (x\right )\right )} \log \left (x - 25\right ) + 15 \, x \]
integrate((((-20*x^5+350*x^4+3320*x^3+10300*x^2+11250*x)*log(x-25)-5*x^5-5 0*x^4-215*x^3-450*x^2-405*x)*log(x)+(-5*x^5+75*x^4+1035*x^3+4925*x^2+10845 *x+10125)*log(x-25)+15*x^2-375*x)/(x^2-25*x),x, algorithm=\
Time = 12.92 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-375 x+15 x^2+\left (10125+10845 x+4925 x^2+1035 x^3+75 x^4-5 x^5\right ) \log (-25+x)+\left (-405 x-450 x^2-215 x^3-50 x^4-5 x^5+\left (11250 x+10300 x^2+3320 x^3+350 x^4-20 x^5\right ) \log (-25+x)\right ) \log (x)}{-25 x+x^2} \, dx=15\,x-\ln \left (x-25\right )\,\ln \left (x\right )\,\left (5\,x^4+50\,x^3+215\,x^2+450\,x+405\right ) \]