Integrand size = 52, antiderivative size = 22 \[ \int \frac {-17860500-12026070 x-2452842 x^2-142884 x^3+\left (16074450+9406530 x+1468530 x^2+39690 x^3\right ) \log (x)+\left (-3572100-1786050 x-198450 x^2\right ) \log ^2(x)}{x^5} \, dx=\frac {3969 (3+x)^2 (-x+5 (-2+\log (x)))^2}{x^4} \] Output:
9*(5*ln(x)-10-x)^2*(63+21*x)^2/x^4
Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(22)=44\).
Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.82 \[ \int \frac {-17860500-12026070 x-2452842 x^2-142884 x^3+\left (16074450+9406530 x+1468530 x^2+39690 x^3\right ) \log (x)+\left (-3572100-1786050 x-198450 x^2\right ) \log ^2(x)}{x^5} \, dx=7938 \left (\frac {450}{x^4}+\frac {390}{x^3}+\frac {229}{2 x^2}+\frac {13}{x}-\frac {450 \log (x)}{x^4}-\frac {345 \log (x)}{x^3}-\frac {80 \log (x)}{x^2}-\frac {5 \log (x)}{x}+\frac {225 \log ^2(x)}{2 x^4}+\frac {75 \log ^2(x)}{x^3}+\frac {25 \log ^2(x)}{2 x^2}\right ) \] Input:
Integrate[(-17860500 - 12026070*x - 2452842*x^2 - 142884*x^3 + (16074450 + 9406530*x + 1468530*x^2 + 39690*x^3)*Log[x] + (-3572100 - 1786050*x - 198 450*x^2)*Log[x]^2)/x^5,x]
Output:
7938*(450/x^4 + 390/x^3 + 229/(2*x^2) + 13/x - (450*Log[x])/x^4 - (345*Log [x])/x^3 - (80*Log[x])/x^2 - (5*Log[x])/x + (225*Log[x]^2)/(2*x^4) + (75*L og[x]^2)/x^3 + (25*Log[x]^2)/(2*x^2))
Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(22)=44\).
Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.45, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, 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.
\(\displaystyle \int \frac {-142884 x^3-2452842 x^2+\left (-198450 x^2-1786050 x-3572100\right ) \log ^2(x)+\left (39690 x^3+1468530 x^2+9406530 x+16074450\right ) \log (x)-12026070 x-17860500}{x^5} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (-\frac {198450 (x+3) (x+6) \log ^2(x)}{x^5}+\frac {39690 (x+3) \left (x^2+34 x+135\right ) \log (x)}{x^5}-\frac {23814 \left (6 x^3+103 x^2+505 x+750\right )}{x^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3572100}{x^4}+\frac {893025 \log ^2(x)}{x^4}-\frac {3572100 \log (x)}{x^4}+\frac {3095820}{x^3}+\frac {595350 \log ^2(x)}{x^3}-\frac {2738610 \log (x)}{x^3}+\frac {908901}{x^2}+\frac {99225 \log ^2(x)}{x^2}-\frac {635040 \log (x)}{x^2}+\frac {103194}{x}-\frac {39690 \log (x)}{x}\) |
Input:
Int[(-17860500 - 12026070*x - 2452842*x^2 - 142884*x^3 + (16074450 + 94065 30*x + 1468530*x^2 + 39690*x^3)*Log[x] + (-3572100 - 1786050*x - 198450*x^ 2)*Log[x]^2)/x^5,x]
Output:
3572100/x^4 + 3095820/x^3 + 908901/x^2 + 103194/x - (3572100*Log[x])/x^4 - (2738610*Log[x])/x^3 - (635040*Log[x])/x^2 - (39690*Log[x])/x + (893025*L og[x]^2)/x^4 + (595350*Log[x]^2)/x^3 + (99225*Log[x]^2)/x^2
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]]
Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(23)=46\).
Time = 0.91 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.68
method | result | size |
risch | \(\frac {99225 \left (x^{2}+6 x +9\right ) \ln \left (x \right )^{2}}{x^{4}}-\frac {39690 \left (x^{3}+16 x^{2}+69 x +90\right ) \ln \left (x \right )}{x^{4}}+\frac {103194 x^{3}+908901 x^{2}+3095820 x +3572100}{x^{4}}\) | \(59\) |
norman | \(\frac {3572100+3095820 x +908901 x^{2}+103194 x^{3}+893025 \ln \left (x \right )^{2}-2738610 x \ln \left (x \right )+595350 x \ln \left (x \right )^{2}-635040 x^{2} \ln \left (x \right )+99225 x^{2} \ln \left (x \right )^{2}-39690 x^{3} \ln \left (x \right )-3572100 \ln \left (x \right )}{x^{4}}\) | \(65\) |
parallelrisch | \(-\frac {-3572100+39690 x^{3} \ln \left (x \right )-99225 x^{2} \ln \left (x \right )^{2}-103194 x^{3}+635040 x^{2} \ln \left (x \right )-595350 x \ln \left (x \right )^{2}-908901 x^{2}+2738610 x \ln \left (x \right )-893025 \ln \left (x \right )^{2}-3095820 x +3572100 \ln \left (x \right )}{x^{4}}\) | \(66\) |
default | \(\frac {99225 \ln \left (x \right )^{2}}{x^{2}}-\frac {635040 \ln \left (x \right )}{x^{2}}+\frac {908901}{x^{2}}-\frac {39690 \ln \left (x \right )}{x}+\frac {103194}{x}+\frac {595350 \ln \left (x \right )^{2}}{x^{3}}-\frac {2738610 \ln \left (x \right )}{x^{3}}+\frac {3095820}{x^{3}}+\frac {893025 \ln \left (x \right )^{2}}{x^{4}}-\frac {3572100 \ln \left (x \right )}{x^{4}}+\frac {3572100}{x^{4}}\) | \(77\) |
parts | \(\frac {99225 \ln \left (x \right )^{2}}{x^{2}}-\frac {635040 \ln \left (x \right )}{x^{2}}+\frac {908901}{x^{2}}-\frac {39690 \ln \left (x \right )}{x}+\frac {103194}{x}+\frac {595350 \ln \left (x \right )^{2}}{x^{3}}-\frac {2738610 \ln \left (x \right )}{x^{3}}+\frac {3095820}{x^{3}}+\frac {893025 \ln \left (x \right )^{2}}{x^{4}}-\frac {3572100 \ln \left (x \right )}{x^{4}}+\frac {3572100}{x^{4}}\) | \(77\) |
orering | \(-\frac {\left (50 x^{8}+325 x^{7}-6679 x^{6}-73233 x^{5}-254529 x^{4}-49059 x^{3}+2253825 x^{2}+6469875 x +5558625\right ) \left (\left (-198450 x^{2}-1786050 x -3572100\right ) \ln \left (x \right )^{2}+\left (39690 x^{3}+1468530 x^{2}+9406530 x +16074450\right ) \ln \left (x \right )-142884 x^{3}-2452842 x^{2}-12026070 x -17860500\right )}{2 x^{4} \left (5 x^{5}-20 x^{4}-874 x^{3}+1500 x^{2}-7875 x +108000\right ) \left (3+x \right )^{3}}-\frac {x^{2} \left (35 x^{7}+135 x^{6}-3897 x^{5}-24399 x^{4}-37881 x^{3}+83835 x^{2}+443475 x +455625\right ) \left (\frac {\left (-396900 x -1786050\right ) \ln \left (x \right )^{2}+\frac {2 \left (-198450 x^{2}-1786050 x -3572100\right ) \ln \left (x \right )}{x}+\left (119070 x^{2}+2937060 x +9406530\right ) \ln \left (x \right )+\frac {39690 x^{3}+1468530 x^{2}+9406530 x +16074450}{x}-428652 x^{2}-4905684 x -12026070}{x^{5}}-\frac {5 \left (\left (-198450 x^{2}-1786050 x -3572100\right ) \ln \left (x \right )^{2}+\left (39690 x^{3}+1468530 x^{2}+9406530 x +16074450\right ) \ln \left (x \right )-142884 x^{3}-2452842 x^{2}-12026070 x -17860500\right )}{x^{6}}\right )}{2 \left (5 x^{5}-20 x^{4}-874 x^{3}+1500 x^{2}-7875 x +108000\right ) \left (3+x \right )^{2}}-\frac {\left (5 x^{6}-433 x^{4}-1011 x^{3}-135 x^{2}+7425 x +10125\right ) x^{3} \left (\frac {-396900 \ln \left (x \right )^{2}+\frac {4 \left (-396900 x -1786050\right ) \ln \left (x \right )}{x}+\frac {-396900 x^{2}-3572100 x -7144200}{x^{2}}-\frac {2 \left (-198450 x^{2}-1786050 x -3572100\right ) \ln \left (x \right )}{x^{2}}+\left (238140 x +2937060\right ) \ln \left (x \right )+\frac {238140 x^{2}+5874120 x +18813060}{x}-\frac {39690 x^{3}+1468530 x^{2}+9406530 x +16074450}{x^{2}}-857304 x -4905684}{x^{5}}-\frac {10 \left (\left (-396900 x -1786050\right ) \ln \left (x \right )^{2}+\frac {2 \left (-198450 x^{2}-1786050 x -3572100\right ) \ln \left (x \right )}{x}+\left (119070 x^{2}+2937060 x +9406530\right ) \ln \left (x \right )+\frac {39690 x^{3}+1468530 x^{2}+9406530 x +16074450}{x}-428652 x^{2}-4905684 x -12026070\right )}{x^{6}}+\frac {30 \left (-198450 x^{2}-1786050 x -3572100\right ) \ln \left (x \right )^{2}+30 \left (39690 x^{3}+1468530 x^{2}+9406530 x +16074450\right ) \ln \left (x \right )-4286520 x^{3}-73585260 x^{2}-360782100 x -535815000}{x^{7}}\right )}{2 \left (5 x^{4}+5 x^{3}-849 x^{2}-2745 x -21600\right ) \left (3+x \right ) \left (-5+x \right )}\) | \(618\) |
Input:
int(((-198450*x^2-1786050*x-3572100)*ln(x)^2+(39690*x^3+1468530*x^2+940653 0*x+16074450)*ln(x)-142884*x^3-2452842*x^2-12026070*x-17860500)/x^5,x,meth od=_RETURNVERBOSE)
Output:
99225*(x^2+6*x+9)/x^4*ln(x)^2-39690*(x^3+16*x^2+69*x+90)/x^4*ln(x)+3969*(2 6*x^3+229*x^2+780*x+900)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {-17860500-12026070 x-2452842 x^2-142884 x^3+\left (16074450+9406530 x+1468530 x^2+39690 x^3\right ) \log (x)+\left (-3572100-1786050 x-198450 x^2\right ) \log ^2(x)}{x^5} \, dx=\frac {3969 \, {\left (26 \, x^{3} + 25 \, {\left (x^{2} + 6 \, x + 9\right )} \log \left (x\right )^{2} + 229 \, x^{2} - 10 \, {\left (x^{3} + 16 \, x^{2} + 69 \, x + 90\right )} \log \left (x\right ) + 780 \, x + 900\right )}}{x^{4}} \] Input:
integrate(((-198450*x^2-1786050*x-3572100)*log(x)^2+(39690*x^3+1468530*x^2 +9406530*x+16074450)*log(x)-142884*x^3-2452842*x^2-12026070*x-17860500)/x^ 5,x, algorithm="fricas")
Output:
3969*(26*x^3 + 25*(x^2 + 6*x + 9)*log(x)^2 + 229*x^2 - 10*(x^3 + 16*x^2 + 69*x + 90)*log(x) + 780*x + 900)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {-17860500-12026070 x-2452842 x^2-142884 x^3+\left (16074450+9406530 x+1468530 x^2+39690 x^3\right ) \log (x)+\left (-3572100-1786050 x-198450 x^2\right ) \log ^2(x)}{x^5} \, dx=\frac {\left (99225 x^{2} + 595350 x + 893025\right ) \log {\left (x \right )}^{2}}{x^{4}} - \frac {- 103194 x^{3} - 908901 x^{2} - 3095820 x - 3572100}{x^{4}} + \frac {\left (- 39690 x^{3} - 635040 x^{2} - 2738610 x - 3572100\right ) \log {\left (x \right )}}{x^{4}} \] Input:
integrate(((-198450*x**2-1786050*x-3572100)*ln(x)**2+(39690*x**3+1468530*x **2+9406530*x+16074450)*ln(x)-142884*x**3-2452842*x**2-12026070*x-17860500 )/x**5,x)
Output:
(99225*x**2 + 595350*x + 893025)*log(x)**2/x**4 - (-103194*x**3 - 908901*x **2 - 3095820*x - 3572100)/x**4 + (-39690*x**3 - 635040*x**2 - 2738610*x - 3572100)*log(x)/x**4
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (19) = 38\).
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.55 \[ \int \frac {-17860500-12026070 x-2452842 x^2-142884 x^3+\left (16074450+9406530 x+1468530 x^2+39690 x^3\right ) \log (x)+\left (-3572100-1786050 x-198450 x^2\right ) \log ^2(x)}{x^5} \, dx=-\frac {39690 \, \log \left (x\right )}{x} + \frac {99225 \, {\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )}}{2 \, x^{2}} + \frac {103194}{x} - \frac {734265 \, \log \left (x\right )}{x^{2}} + \frac {66150 \, {\left (9 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) + 2\right )}}{x^{3}} + \frac {1718577}{2 \, x^{2}} - \frac {3135510 \, \log \left (x\right )}{x^{3}} + \frac {893025 \, {\left (8 \, \log \left (x\right )^{2} + 4 \, \log \left (x\right ) + 1\right )}}{8 \, x^{4}} + \frac {2963520}{x^{3}} - \frac {8037225 \, \log \left (x\right )}{2 \, x^{4}} + \frac {27683775}{8 \, x^{4}} \] Input:
integrate(((-198450*x^2-1786050*x-3572100)*log(x)^2+(39690*x^3+1468530*x^2 +9406530*x+16074450)*log(x)-142884*x^3-2452842*x^2-12026070*x-17860500)/x^ 5,x, algorithm="maxima")
Output:
-39690*log(x)/x + 99225/2*(2*log(x)^2 + 2*log(x) + 1)/x^2 + 103194/x - 734 265*log(x)/x^2 + 66150*(9*log(x)^2 + 6*log(x) + 2)/x^3 + 1718577/2/x^2 - 3 135510*log(x)/x^3 + 893025/8*(8*log(x)^2 + 4*log(x) + 1)/x^4 + 2963520/x^3 - 8037225/2*log(x)/x^4 + 27683775/8/x^4
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (19) = 38\).
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64 \[ \int \frac {-17860500-12026070 x-2452842 x^2-142884 x^3+\left (16074450+9406530 x+1468530 x^2+39690 x^3\right ) \log (x)+\left (-3572100-1786050 x-198450 x^2\right ) \log ^2(x)}{x^5} \, dx=\frac {99225 \, {\left (x^{2} + 6 \, x + 9\right )} \log \left (x\right )^{2}}{x^{4}} - \frac {39690 \, {\left (x^{3} + 16 \, x^{2} + 69 \, x + 90\right )} \log \left (x\right )}{x^{4}} + \frac {3969 \, {\left (26 \, x^{3} + 229 \, x^{2} + 780 \, x + 900\right )}}{x^{4}} \] Input:
integrate(((-198450*x^2-1786050*x-3572100)*log(x)^2+(39690*x^3+1468530*x^2 +9406530*x+16074450)*log(x)-142884*x^3-2452842*x^2-12026070*x-17860500)/x^ 5,x, algorithm="giac")
Output:
99225*(x^2 + 6*x + 9)*log(x)^2/x^4 - 39690*(x^3 + 16*x^2 + 69*x + 90)*log( x)/x^4 + 3969*(26*x^3 + 229*x^2 + 780*x + 900)/x^4
Time = 1.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \frac {-17860500-12026070 x-2452842 x^2-142884 x^3+\left (16074450+9406530 x+1468530 x^2+39690 x^3\right ) \log (x)+\left (-3572100-1786050 x-198450 x^2\right ) \log ^2(x)}{x^5} \, dx=\frac {x\,\left (893025\,{\ln \left (x\right )}^2-3572100\,\ln \left (x\right )+3572100\right )+x^3\,\left (99225\,{\ln \left (x\right )}^2-635040\,\ln \left (x\right )+908901\right )+x^2\,\left (595350\,{\ln \left (x\right )}^2-2738610\,\ln \left (x\right )+3095820\right )-x^4\,\left (39690\,\ln \left (x\right )-103194\right )}{x^5} \] Input:
int(-(12026070*x + log(x)^2*(1786050*x + 198450*x^2 + 3572100) + 2452842*x ^2 + 142884*x^3 - log(x)*(9406530*x + 1468530*x^2 + 39690*x^3 + 16074450) + 17860500)/x^5,x)
Output:
(x*(893025*log(x)^2 - 3572100*log(x) + 3572100) + x^3*(99225*log(x)^2 - 63 5040*log(x) + 908901) + x^2*(595350*log(x)^2 - 2738610*log(x) + 3095820) - x^4*(39690*log(x) - 103194))/x^5
Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.95 \[ \int \frac {-17860500-12026070 x-2452842 x^2-142884 x^3+\left (16074450+9406530 x+1468530 x^2+39690 x^3\right ) \log (x)+\left (-3572100-1786050 x-198450 x^2\right ) \log ^2(x)}{x^5} \, dx=\frac {99225 \mathrm {log}\left (x \right )^{2} x^{2}+595350 \mathrm {log}\left (x \right )^{2} x +893025 \mathrm {log}\left (x \right )^{2}-39690 \,\mathrm {log}\left (x \right ) x^{3}-635040 \,\mathrm {log}\left (x \right ) x^{2}-2738610 \,\mathrm {log}\left (x \right ) x -3572100 \,\mathrm {log}\left (x \right )+103194 x^{3}+908901 x^{2}+3095820 x +3572100}{x^{4}} \] Input:
int(((-198450*x^2-1786050*x-3572100)*log(x)^2+(39690*x^3+1468530*x^2+94065 30*x+16074450)*log(x)-142884*x^3-2452842*x^2-12026070*x-17860500)/x^5,x)
Output:
(3969*(25*log(x)**2*x**2 + 150*log(x)**2*x + 225*log(x)**2 - 10*log(x)*x** 3 - 160*log(x)*x**2 - 690*log(x)*x - 900*log(x) + 26*x**3 + 229*x**2 + 780 *x + 900))/x**4