Integrand size = 82, antiderivative size = 24 \[ \int \frac {-80-800 x-3320 x^2-7400 x^3-9605 x^4-7360 x^5-3320 x^6-800 x^7-80 x^8+\left (400+3200 x+9960 x^2+14800 x^3+9605 x^4-3320 x^6-1600 x^7-240 x^8\right ) \log (x)}{x^6} \, dx=5 \left (9-\frac {x+\left (5+2 \left (\frac {1}{x}+x\right )\right )^4}{x}\right ) \log (x) \] Output:
5*(9-(x+(2*x+2/x+5)^4)/x)*ln(x)
Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(24)=48\).
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46 \[ \int \frac {-80-800 x-3320 x^2-7400 x^3-9605 x^4-7360 x^5-3320 x^6-800 x^7-80 x^8+\left (400+3200 x+9960 x^2+14800 x^3+9605 x^4-3320 x^6-1600 x^7-240 x^8\right ) \log (x)}{x^6} \, dx=-7360 \log (x)-\frac {80 \log (x)}{x^5}-\frac {800 \log (x)}{x^4}-\frac {3320 \log (x)}{x^3}-\frac {7400 \log (x)}{x^2}-\frac {9605 \log (x)}{x}-3320 x \log (x)-800 x^2 \log (x)-80 x^3 \log (x) \] Input:
Integrate[(-80 - 800*x - 3320*x^2 - 7400*x^3 - 9605*x^4 - 7360*x^5 - 3320* x^6 - 800*x^7 - 80*x^8 + (400 + 3200*x + 9960*x^2 + 14800*x^3 + 9605*x^4 - 3320*x^6 - 1600*x^7 - 240*x^8)*Log[x])/x^6,x]
Output:
-7360*Log[x] - (80*Log[x])/x^5 - (800*Log[x])/x^4 - (3320*Log[x])/x^3 - (7 400*Log[x])/x^2 - (9605*Log[x])/x - 3320*x*Log[x] - 800*x^2*Log[x] - 80*x^ 3*Log[x]
Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(24)=48\).
Time = 0.45 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2010, 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.
\(\displaystyle \int \frac {-80 x^8-800 x^7-3320 x^6-7360 x^5-9605 x^4-7400 x^3-3320 x^2+\left (-240 x^8-1600 x^7-3320 x^6+9605 x^4+14800 x^3+9960 x^2+3200 x+400\right ) \log (x)-800 x-80}{x^6} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (-\frac {5 (x+2)^3 \left (6 x^2-5 x-10\right ) (2 x+1)^3 \log (x)}{x^6}-\frac {5 \left (16 x^8+160 x^7+664 x^6+1472 x^5+1921 x^4+1480 x^3+664 x^2+160 x+16\right )}{x^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {80 \log (x)}{x^5}-\frac {800 \log (x)}{x^4}-80 x^3 \log (x)-\frac {3320 \log (x)}{x^3}-800 x^2 \log (x)-\frac {7400 \log (x)}{x^2}-3320 x \log (x)-7360 \log (x)-\frac {9605 \log (x)}{x}\) |
Input:
Int[(-80 - 800*x - 3320*x^2 - 7400*x^3 - 9605*x^4 - 7360*x^5 - 3320*x^6 - 800*x^7 - 80*x^8 + (400 + 3200*x + 9960*x^2 + 14800*x^3 + 9605*x^4 - 3320* x^6 - 1600*x^7 - 240*x^8)*Log[x])/x^6,x]
Output:
-7360*Log[x] - (80*Log[x])/x^5 - (800*Log[x])/x^4 - (3320*Log[x])/x^3 - (7 400*Log[x])/x^2 - (9605*Log[x])/x - 3320*x*Log[x] - 800*x^2*Log[x] - 80*x^ 3*Log[x]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 2.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00
method | result | size |
risch | \(-\frac {5 \left (16 x^{8}+160 x^{7}+664 x^{6}+1921 x^{4}+1480 x^{3}+664 x^{2}+160 x +16\right ) \ln \left (x \right )}{x^{5}}-7360 \ln \left (x \right )\) | \(48\) |
default | \(-\frac {3320 \ln \left (x \right )}{x^{3}}-\frac {7400 \ln \left (x \right )}{x^{2}}-80 x^{3} \ln \left (x \right )-3320 x \ln \left (x \right )-800 x^{2} \ln \left (x \right )-\frac {800 \ln \left (x \right )}{x^{4}}-\frac {80 \ln \left (x \right )}{x^{5}}-\frac {9605 \ln \left (x \right )}{x}-7360 \ln \left (x \right )\) | \(60\) |
parts | \(-\frac {3320 \ln \left (x \right )}{x^{3}}-\frac {7400 \ln \left (x \right )}{x^{2}}-80 x^{3} \ln \left (x \right )-3320 x \ln \left (x \right )-800 x^{2} \ln \left (x \right )-\frac {800 \ln \left (x \right )}{x^{4}}-\frac {80 \ln \left (x \right )}{x^{5}}-\frac {9605 \ln \left (x \right )}{x}-7360 \ln \left (x \right )\) | \(60\) |
norman | \(\frac {-7360 x^{5} \ln \left (x \right )-800 x \ln \left (x \right )-3320 x^{2} \ln \left (x \right )-7400 x^{3} \ln \left (x \right )-9605 x^{4} \ln \left (x \right )-800 x^{7} \ln \left (x \right )-3320 \ln \left (x \right ) x^{6}-80 \ln \left (x \right ) x^{8}-80 \ln \left (x \right )}{x^{5}}\) | \(64\) |
parallelrisch | \(-\frac {80 \ln \left (x \right ) x^{8}+800 x^{7} \ln \left (x \right )+3320 \ln \left (x \right ) x^{6}+7360 x^{5} \ln \left (x \right )+9605 x^{4} \ln \left (x \right )+7400 x^{3} \ln \left (x \right )+3320 x^{2} \ln \left (x \right )+800 x \ln \left (x \right )+80 \ln \left (x \right )}{x^{5}}\) | \(65\) |
orering | \(\frac {\left (1280 x^{16}+20480 x^{15}+140544 x^{14}-757152 x^{12}-5212720 x^{10}-36454464 x^{9}-74965635 x^{8}-74525248 x^{7}-43474480 x^{6}-18359424 x^{5}-6831776 x^{4}-2078720 x^{3}-421632 x^{2}-51200 x -2816\right ) \left (\left (-240 x^{8}-1600 x^{7}-3320 x^{6}+9605 x^{4}+14800 x^{3}+9960 x^{2}+3200 x +400\right ) \ln \left (x \right )-80 x^{8}-800 x^{7}-3320 x^{6}-7360 x^{5}-9605 x^{4}-7400 x^{3}-3320 x^{2}-800 x -80\right )}{x^{5} \left (2 x^{2}+5 x +2\right ) \left (2+x \right ) \left (1+2 x \right ) \left (576 x^{12}+4160 x^{11}+5360 x^{10}-37312 x^{9}-173380 x^{8}-318548 x^{7}-273215 x^{6}-31900 x^{5}+159900 x^{4}+160800 x^{3}+73456 x^{2}+16960 x +1600\right )}-\frac {\left (256 x^{16}+5120 x^{15}+46848 x^{14}-757152 x^{12}-974016 x^{11}+5212720 x^{10}+18227232 x^{9}+24988545 x^{8}+18631312 x^{7}+8694896 x^{6}+3059904 x^{5}+975968 x^{4}+259840 x^{3}+46848 x^{2}+5120 x +256\right ) x^{2} \left (\frac {\left (-1920 x^{7}-11200 x^{6}-19920 x^{5}+38420 x^{3}+44400 x^{2}+19920 x +3200\right ) \ln \left (x \right )+\frac {-240 x^{8}-1600 x^{7}-3320 x^{6}+9605 x^{4}+14800 x^{3}+9960 x^{2}+3200 x +400}{x}-640 x^{7}-5600 x^{6}-19920 x^{5}-36800 x^{4}-38420 x^{3}-22200 x^{2}-6640 x -800}{x^{6}}-\frac {6 \left (\left (-240 x^{8}-1600 x^{7}-3320 x^{6}+9605 x^{4}+14800 x^{3}+9960 x^{2}+3200 x +400\right ) \ln \left (x \right )-80 x^{8}-800 x^{7}-3320 x^{6}-7360 x^{5}-9605 x^{4}-7400 x^{3}-3320 x^{2}-800 x -80\right )}{x^{7}}\right )}{\left (2+x \right )^{2} \left (576 x^{12}+4160 x^{11}+5360 x^{10}-37312 x^{9}-173380 x^{8}-318548 x^{7}-273215 x^{6}-31900 x^{5}+159900 x^{4}+160800 x^{3}+73456 x^{2}+16960 x +1600\right ) \left (1+2 x \right )^{2}}\) | \(589\) |
Input:
int(((-240*x^8-1600*x^7-3320*x^6+9605*x^4+14800*x^3+9960*x^2+3200*x+400)*l n(x)-80*x^8-800*x^7-3320*x^6-7360*x^5-9605*x^4-7400*x^3-3320*x^2-800*x-80) /x^6,x,method=_RETURNVERBOSE)
Output:
-5*(16*x^8+160*x^7+664*x^6+1921*x^4+1480*x^3+664*x^2+160*x+16)/x^5*ln(x)-7 360*ln(x)
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {-80-800 x-3320 x^2-7400 x^3-9605 x^4-7360 x^5-3320 x^6-800 x^7-80 x^8+\left (400+3200 x+9960 x^2+14800 x^3+9605 x^4-3320 x^6-1600 x^7-240 x^8\right ) \log (x)}{x^6} \, dx=-\frac {5 \, {\left (16 \, x^{8} + 160 \, x^{7} + 664 \, x^{6} + 1472 \, x^{5} + 1921 \, x^{4} + 1480 \, x^{3} + 664 \, x^{2} + 160 \, x + 16\right )} \log \left (x\right )}{x^{5}} \] Input:
integrate(((-240*x^8-1600*x^7-3320*x^6+9605*x^4+14800*x^3+9960*x^2+3200*x+ 400)*log(x)-80*x^8-800*x^7-3320*x^6-7360*x^5-9605*x^4-7400*x^3-3320*x^2-80 0*x-80)/x^6,x, algorithm="fricas")
Output:
-5*(16*x^8 + 160*x^7 + 664*x^6 + 1472*x^5 + 1921*x^4 + 1480*x^3 + 664*x^2 + 160*x + 16)*log(x)/x^5
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {-80-800 x-3320 x^2-7400 x^3-9605 x^4-7360 x^5-3320 x^6-800 x^7-80 x^8+\left (400+3200 x+9960 x^2+14800 x^3+9605 x^4-3320 x^6-1600 x^7-240 x^8\right ) \log (x)}{x^6} \, dx=- 7360 \log {\left (x \right )} + \frac {\left (- 80 x^{8} - 800 x^{7} - 3320 x^{6} - 9605 x^{4} - 7400 x^{3} - 3320 x^{2} - 800 x - 80\right ) \log {\left (x \right )}}{x^{5}} \] Input:
integrate(((-240*x**8-1600*x**7-3320*x**6+9605*x**4+14800*x**3+9960*x**2+3 200*x+400)*ln(x)-80*x**8-800*x**7-3320*x**6-7360*x**5-9605*x**4-7400*x**3- 3320*x**2-800*x-80)/x**6,x)
Output:
-7360*log(x) + (-80*x**8 - 800*x**7 - 3320*x**6 - 9605*x**4 - 7400*x**3 - 3320*x**2 - 800*x - 80)*log(x)/x**5
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (24) = 48\).
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46 \[ \int \frac {-80-800 x-3320 x^2-7400 x^3-9605 x^4-7360 x^5-3320 x^6-800 x^7-80 x^8+\left (400+3200 x+9960 x^2+14800 x^3+9605 x^4-3320 x^6-1600 x^7-240 x^8\right ) \log (x)}{x^6} \, dx=-80 \, x^{3} \log \left (x\right ) - 800 \, x^{2} \log \left (x\right ) - 3320 \, x \log \left (x\right ) - \frac {9605 \, \log \left (x\right )}{x} - \frac {7400 \, \log \left (x\right )}{x^{2}} - \frac {3320 \, \log \left (x\right )}{x^{3}} - \frac {800 \, \log \left (x\right )}{x^{4}} - \frac {80 \, \log \left (x\right )}{x^{5}} - 7360 \, \log \left (x\right ) \] Input:
integrate(((-240*x^8-1600*x^7-3320*x^6+9605*x^4+14800*x^3+9960*x^2+3200*x+ 400)*log(x)-80*x^8-800*x^7-3320*x^6-7360*x^5-9605*x^4-7400*x^3-3320*x^2-80 0*x-80)/x^6,x, algorithm="maxima")
Output:
-80*x^3*log(x) - 800*x^2*log(x) - 3320*x*log(x) - 9605*log(x)/x - 7400*log (x)/x^2 - 3320*log(x)/x^3 - 800*log(x)/x^4 - 80*log(x)/x^5 - 7360*log(x)
Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {-80-800 x-3320 x^2-7400 x^3-9605 x^4-7360 x^5-3320 x^6-800 x^7-80 x^8+\left (400+3200 x+9960 x^2+14800 x^3+9605 x^4-3320 x^6-1600 x^7-240 x^8\right ) \log (x)}{x^6} \, dx=-5 \, {\left (16 \, x^{3} + 160 \, x^{2} + 664 \, x + \frac {1921 \, x^{4} + 1480 \, x^{3} + 664 \, x^{2} + 160 \, x + 16}{x^{5}}\right )} \log \left (x\right ) - 7360 \, \log \left (x\right ) \] Input:
integrate(((-240*x^8-1600*x^7-3320*x^6+9605*x^4+14800*x^3+9960*x^2+3200*x+ 400)*log(x)-80*x^8-800*x^7-3320*x^6-7360*x^5-9605*x^4-7400*x^3-3320*x^2-80 0*x-80)/x^6,x, algorithm="giac")
Output:
-5*(16*x^3 + 160*x^2 + 664*x + (1921*x^4 + 1480*x^3 + 664*x^2 + 160*x + 16 )/x^5)*log(x) - 7360*log(x)
Time = 4.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.79 \[ \int \frac {-80-800 x-3320 x^2-7400 x^3-9605 x^4-7360 x^5-3320 x^6-800 x^7-80 x^8+\left (400+3200 x+9960 x^2+14800 x^3+9605 x^4-3320 x^6-1600 x^7-240 x^8\right ) \log (x)}{x^6} \, dx=-\frac {800\,x^2\,\ln \left (x\right )+3320\,x^3\,\ln \left (x\right )+7400\,x^4\,\ln \left (x\right )+9605\,x^5\,\ln \left (x\right )+7360\,x^6\,\ln \left (x\right )+3320\,x^7\,\ln \left (x\right )+800\,x^8\,\ln \left (x\right )+80\,x^9\,\ln \left (x\right )+80\,x\,\ln \left (x\right )}{x^6} \] Input:
int(-(800*x - log(x)*(3200*x + 9960*x^2 + 14800*x^3 + 9605*x^4 - 3320*x^6 - 1600*x^7 - 240*x^8 + 400) + 3320*x^2 + 7400*x^3 + 9605*x^4 + 7360*x^5 + 3320*x^6 + 800*x^7 + 80*x^8 + 80)/x^6,x)
Output:
-(800*x^2*log(x) + 3320*x^3*log(x) + 7400*x^4*log(x) + 9605*x^5*log(x) + 7 360*x^6*log(x) + 3320*x^7*log(x) + 800*x^8*log(x) + 80*x^9*log(x) + 80*x*l og(x))/x^6
Time = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {-80-800 x-3320 x^2-7400 x^3-9605 x^4-7360 x^5-3320 x^6-800 x^7-80 x^8+\left (400+3200 x+9960 x^2+14800 x^3+9605 x^4-3320 x^6-1600 x^7-240 x^8\right ) \log (x)}{x^6} \, dx=\frac {5 \,\mathrm {log}\left (x \right ) \left (-16 x^{8}-160 x^{7}-664 x^{6}-1472 x^{5}-1921 x^{4}-1480 x^{3}-664 x^{2}-160 x -16\right )}{x^{5}} \] Input:
int(((-240*x^8-1600*x^7-3320*x^6+9605*x^4+14800*x^3+9960*x^2+3200*x+400)*l og(x)-80*x^8-800*x^7-3320*x^6-7360*x^5-9605*x^4-7400*x^3-3320*x^2-800*x-80 )/x^6,x)
Output:
(5*log(x)*( - 16*x**8 - 160*x**7 - 664*x**6 - 1472*x**5 - 1921*x**4 - 1480 *x**3 - 664*x**2 - 160*x - 16))/x**5