Integrand size = 37, antiderivative size = 22 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=5+\frac {34225 (5-2 x)^2 (2-\log (x))^2}{x^2} \] Output:
5+34225*(5-2*x)^2*(-ln(x)+2)^2/x^2
Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=\frac {34225 (5-2 x)^2 (-2+\log (x))^2}{x^2} \] Input:
Integrate[(-10267500 + 5476000*x - 547600*x^2 + (8556250 - 4107000*x + 273 800*x^2)*Log[x] + (-1711250 + 684500*x)*Log[x]^2)/x^3,x]
Output:
(34225*(5 - 2*x)^2*(-2 + Log[x])^2)/x^2
Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(22)=44\).
Time = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, 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.
\(\displaystyle \int \frac {-547600 x^2+\left (273800 x^2-4107000 x+8556250\right ) \log (x)+5476000 x+(684500 x-1711250) \log ^2(x)-10267500}{x^3} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {342250 (2 x-5) \log ^2(x)}{x^3}+\frac {68450 (2 x-25) (2 x-5) \log (x)}{x^3}-\frac {136900 \left (4 x^2-40 x+75\right )}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3422500}{x^2}+\frac {855625 \log ^2(x)}{x^2}-\frac {3422500 \log (x)}{x^2}-\frac {2738000}{x}-\frac {684500 \log ^2(x)}{x}+136900 \log ^2(x)+\frac {2738000 \log (x)}{x}-547600 \log (x)\) |
Input:
Int[(-10267500 + 5476000*x - 547600*x^2 + (8556250 - 4107000*x + 273800*x^ 2)*Log[x] + (-1711250 + 684500*x)*Log[x]^2)/x^3,x]
Output:
3422500/x^2 - 2738000/x - 547600*Log[x] - (3422500*Log[x])/x^2 + (2738000* Log[x])/x + 136900*Log[x]^2 + (855625*Log[x]^2)/x^2 - (684500*Log[x]^2)/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]]
Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).
Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18
method | result | size |
norman | \(\frac {3422500-547600 x^{2} \ln \left (x \right )-2738000 x +855625 \ln \left (x \right )^{2}+2738000 x \ln \left (x \right )-684500 x \ln \left (x \right )^{2}+136900 x^{2} \ln \left (x \right )^{2}-3422500 \ln \left (x \right )}{x^{2}}\) | \(48\) |
risch | \(\frac {34225 \left (4 x^{2}-20 x +25\right ) \ln \left (x \right )^{2}}{x^{2}}+\frac {684500 \left (-5+4 x \right ) \ln \left (x \right )}{x^{2}}-\frac {136900 \left (4 x^{2} \ln \left (x \right )+20 x -25\right )}{x^{2}}\) | \(50\) |
default | \(-\frac {684500 \ln \left (x \right )^{2}}{x}+\frac {2738000 \ln \left (x \right )}{x}-\frac {2738000}{x}+136900 \ln \left (x \right )^{2}+\frac {855625 \ln \left (x \right )^{2}}{x^{2}}-\frac {3422500 \ln \left (x \right )}{x^{2}}+\frac {3422500}{x^{2}}-547600 \ln \left (x \right )\) | \(54\) |
parts | \(-\frac {684500 \ln \left (x \right )^{2}}{x}+\frac {2738000 \ln \left (x \right )}{x}-\frac {2738000}{x}+136900 \ln \left (x \right )^{2}+\frac {855625 \ln \left (x \right )^{2}}{x^{2}}-\frac {3422500 \ln \left (x \right )}{x^{2}}+\frac {3422500}{x^{2}}-547600 \ln \left (x \right )\) | \(54\) |
orering | \(\frac {\left (64 x^{6}+320 x^{5}+16400 x^{4}-156000 x^{3}+487500 x^{2}-487500 x +296875\right ) \left (\left (684500 x -1711250\right ) \ln \left (x \right )^{2}+\left (273800 x^{2}-4107000 x +8556250\right ) \ln \left (x \right )-547600 x^{2}+5476000 x -10267500\right )}{20 x^{2} \left (-5+2 x \right )^{3} \left (2 x^{2}+35 x +50\right )}+\frac {\left (96 x^{5}-400 x^{4}+14000 x^{3}-61000 x^{2}+48750 x -28125\right ) x^{2} \left (\frac {684500 \ln \left (x \right )^{2}+\frac {2 \left (684500 x -1711250\right ) \ln \left (x \right )}{x}+\left (547600 x -4107000\right ) \ln \left (x \right )+\frac {273800 x^{2}-4107000 x +8556250}{x}-1095200 x +5476000}{x^{3}}-\frac {3 \left (\left (684500 x -1711250\right ) \ln \left (x \right )^{2}+\left (273800 x^{2}-4107000 x +8556250\right ) \ln \left (x \right )-547600 x^{2}+5476000 x -10267500\right )}{x^{4}}\right )}{20 \left (-5+2 x \right )^{2} \left (2 x^{2}+35 x +50\right )}+\frac {\left (16 x^{4}+1400 x^{2}-1000 x +625\right ) x^{3} \left (\frac {\frac {2738000 \ln \left (x \right )}{x}+\frac {1369000 x -3422500}{x^{2}}-\frac {2 \left (684500 x -1711250\right ) \ln \left (x \right )}{x^{2}}+547600 \ln \left (x \right )+\frac {1095200 x -8214000}{x}-\frac {273800 x^{2}-4107000 x +8556250}{x^{2}}-1095200}{x^{3}}-\frac {6 \left (684500 \ln \left (x \right )^{2}+\frac {2 \left (684500 x -1711250\right ) \ln \left (x \right )}{x}+\left (547600 x -4107000\right ) \ln \left (x \right )+\frac {273800 x^{2}-4107000 x +8556250}{x}-1095200 x +5476000\right )}{x^{4}}+\frac {12 \left (684500 x -1711250\right ) \ln \left (x \right )^{2}+12 \left (273800 x^{2}-4107000 x +8556250\right ) \ln \left (x \right )-6571200 x^{2}+65712000 x -123210000}{x^{5}}\right )}{20 \left (2 x^{2}+35 x +50\right ) \left (-5+2 x \right )}\) | \(418\) |
Input:
int(((684500*x-1711250)*ln(x)^2+(273800*x^2-4107000*x+8556250)*ln(x)-54760 0*x^2+5476000*x-10267500)/x^3,x,method=_RETURNVERBOSE)
Output:
(3422500-547600*x^2*ln(x)-2738000*x+855625*ln(x)^2+2738000*x*ln(x)-684500* x*ln(x)^2+136900*x^2*ln(x)^2-3422500*ln(x))/x^2
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=\frac {34225 \, {\left ({\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (x\right )^{2} - 4 \, {\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (x\right ) - 80 \, x + 100\right )}}{x^{2}} \] Input:
integrate(((684500*x-1711250)*log(x)^2+(273800*x^2-4107000*x+8556250)*log( x)-547600*x^2+5476000*x-10267500)/x^3,x, algorithm="fricas")
Output:
34225*((4*x^2 - 20*x + 25)*log(x)^2 - 4*(4*x^2 - 20*x + 25)*log(x) - 80*x + 100)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=- 547600 \log {\left (x \right )} + \frac {\left (2738000 x - 3422500\right ) \log {\left (x \right )}}{x^{2}} - \frac {2738000 x - 3422500}{x^{2}} + \frac {\left (136900 x^{2} - 684500 x + 855625\right ) \log {\left (x \right )}^{2}}{x^{2}} \] Input:
integrate(((684500*x-1711250)*ln(x)**2+(273800*x**2-4107000*x+8556250)*ln( x)-547600*x**2+5476000*x-10267500)/x**3,x)
Output:
-547600*log(x) + (2738000*x - 3422500)*log(x)/x**2 - (2738000*x - 3422500) /x**2 + (136900*x**2 - 684500*x + 855625)*log(x)**2/x**2
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (20) = 40\).
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.05 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=136900 \, \log \left (x\right )^{2} - \frac {684500 \, {\left (\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 2\right )}}{x} + \frac {4107000 \, \log \left (x\right )}{x} + \frac {855625 \, {\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )}}{2 \, x^{2}} - \frac {1369000}{x} - \frac {4278125 \, \log \left (x\right )}{x^{2}} + \frac {5989375}{2 \, x^{2}} - 547600 \, \log \left (x\right ) \] Input:
integrate(((684500*x-1711250)*log(x)^2+(273800*x^2-4107000*x+8556250)*log( x)-547600*x^2+5476000*x-10267500)/x^3,x, algorithm="maxima")
Output:
136900*log(x)^2 - 684500*(log(x)^2 + 2*log(x) + 2)/x + 4107000*log(x)/x + 855625/2*(2*log(x)^2 + 2*log(x) + 1)/x^2 - 1369000/x - 4278125*log(x)/x^2 + 5989375/2/x^2 - 547600*log(x)
\[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=\int { \frac {68450 \, {\left (5 \, {\left (2 \, x - 5\right )} \log \left (x\right )^{2} - 8 \, x^{2} + {\left (4 \, x^{2} - 60 \, x + 125\right )} \log \left (x\right ) + 80 \, x - 150\right )}}{x^{3}} \,d x } \] Input:
integrate(((684500*x-1711250)*log(x)^2+(273800*x^2-4107000*x+8556250)*log( x)-547600*x^2+5476000*x-10267500)/x^3,x, algorithm="giac")
Output:
integrate(68450*(5*(2*x - 5)*log(x)^2 - 8*x^2 + (4*x^2 - 60*x + 125)*log(x ) + 80*x - 150)/x^3, x)
Time = 2.96 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=\frac {x\,\left (855625\,{\ln \left (x\right )}^2-3422500\,\ln \left (x\right )+3422500\right )-x^2\,\left (684500\,{\ln \left (x\right )}^2-2738000\,\ln \left (x\right )+2738000\right )}{x^3}-547600\,\ln \left (x\right )+136900\,{\ln \left (x\right )}^2 \] Input:
int((5476000*x + log(x)*(273800*x^2 - 4107000*x + 8556250) - 547600*x^2 + log(x)^2*(684500*x - 1711250) - 10267500)/x^3,x)
Output:
(x*(855625*log(x)^2 - 3422500*log(x) + 3422500) - x^2*(684500*log(x)^2 - 2 738000*log(x) + 2738000))/x^3 - 547600*log(x) + 136900*log(x)^2
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=\frac {136900 \mathrm {log}\left (x \right )^{2} x^{2}-684500 \mathrm {log}\left (x \right )^{2} x +855625 \mathrm {log}\left (x \right )^{2}-547600 \,\mathrm {log}\left (x \right ) x^{2}+2738000 \,\mathrm {log}\left (x \right ) x -3422500 \,\mathrm {log}\left (x \right )-2738000 x +3422500}{x^{2}} \] Input:
int(((684500*x-1711250)*log(x)^2+(273800*x^2-4107000*x+8556250)*log(x)-547 600*x^2+5476000*x-10267500)/x^3,x)
Output:
(34225*(4*log(x)**2*x**2 - 20*log(x)**2*x + 25*log(x)**2 - 16*log(x)*x**2 + 80*log(x)*x - 100*log(x) - 80*x + 100))/x**2