Integrand size = 58, antiderivative size = 23 \[ \int \frac {-240 x^3-48 x^4+40 x^6+\left (3600+1440 x+144 x^2-480 x^3-144 x^4\right ) \log (x)+\left (-1800+72 x^2\right ) \log ^2(x)}{5 x^2} \, dx=\frac {2}{5} x \left (-2 x^2+\frac {6 (5+x) \log (x)}{x}\right )^2 \] Output:
2/5*(6*(5+x)/x*ln(x)-2*x^2)^2*x
Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(23)=46\).
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {-240 x^3-48 x^4+40 x^6+\left (3600+1440 x+144 x^2-480 x^3-144 x^4\right ) \log (x)+\left (-1800+72 x^2\right ) \log ^2(x)}{5 x^2} \, dx=\frac {8 x^5}{5}-48 x^2 \log (x)-\frac {48}{5} x^3 \log (x)+144 \log ^2(x)+\frac {360 \log ^2(x)}{x}+\frac {72}{5} x \log ^2(x) \] Input:
Integrate[(-240*x^3 - 48*x^4 + 40*x^6 + (3600 + 1440*x + 144*x^2 - 480*x^3 - 144*x^4)*Log[x] + (-1800 + 72*x^2)*Log[x]^2)/(5*x^2),x]
Output:
(8*x^5)/5 - 48*x^2*Log[x] - (48*x^3*Log[x])/5 + 144*Log[x]^2 + (360*Log[x] ^2)/x + (72*x*Log[x]^2)/5
Time = 0.37 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 27, 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.
\(\displaystyle \int \frac {40 x^6-48 x^4-240 x^3+\left (72 x^2-1800\right ) \log ^2(x)+\left (-144 x^4-480 x^3+144 x^2+1440 x+3600\right ) \log (x)}{5 x^2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int -\frac {8 \left (-5 x^6+6 x^4+30 x^3+9 \left (25-x^2\right ) \log ^2(x)-6 \left (-3 x^4-10 x^3+3 x^2+30 x+75\right ) \log (x)\right )}{x^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {8}{5} \int \frac {-5 x^6+6 x^4+30 x^3+9 \left (25-x^2\right ) \log ^2(x)-6 \left (-3 x^4-10 x^3+3 x^2+30 x+75\right ) \log (x)}{x^2}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle -\frac {8}{5} \int \left (-\frac {9 (x-5) (x+5) \log ^2(x)}{x^2}+\frac {6 \left (3 x^4+10 x^3-3 x^2-30 x-75\right ) \log (x)}{x^2}-x \left (5 x^3-6 x-30\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {8}{5} \left (-x^5+6 x^3 \log (x)+30 x^2 \log (x)-9 x \log ^2(x)-90 \log ^2(x)-\frac {225 \log ^2(x)}{x}\right )\) |
Input:
Int[(-240*x^3 - 48*x^4 + 40*x^6 + (3600 + 1440*x + 144*x^2 - 480*x^3 - 144 *x^4)*Log[x] + (-1800 + 72*x^2)*Log[x]^2)/(5*x^2),x]
Output:
(-8*(-x^5 + 30*x^2*Log[x] + 6*x^3*Log[x] - 90*Log[x]^2 - (225*Log[x]^2)/x - 9*x*Log[x]^2))/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 = 1.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70
method | result | size |
risch | \(\frac {72 \left (x^{2}+10 x +25\right ) \ln \left (x \right )^{2}}{5 x}+\frac {\left (-48 x^{3}-240 x^{2}\right ) \ln \left (x \right )}{5}+\frac {8 x^{5}}{5}\) | \(39\) |
default | \(\frac {8 x^{5}}{5}-\frac {48 x^{3} \ln \left (x \right )}{5}+\frac {72 x \ln \left (x \right )^{2}}{5}-48 x^{2} \ln \left (x \right )+\frac {360 \ln \left (x \right )^{2}}{x}+144 \ln \left (x \right )^{2}\) | \(43\) |
parts | \(\frac {8 x^{5}}{5}-\frac {48 x^{3} \ln \left (x \right )}{5}+\frac {72 x \ln \left (x \right )^{2}}{5}-48 x^{2} \ln \left (x \right )+\frac {360 \ln \left (x \right )^{2}}{x}+144 \ln \left (x \right )^{2}\) | \(43\) |
norman | \(\frac {\frac {8 x^{6}}{5}+360 \ln \left (x \right )^{2}+144 x \ln \left (x \right )^{2}+\frac {72 x^{2} \ln \left (x \right )^{2}}{5}-48 x^{3} \ln \left (x \right )-\frac {48 x^{4} \ln \left (x \right )}{5}}{x}\) | \(47\) |
parallelrisch | \(-\frac {-8 x^{6}+48 x^{4} \ln \left (x \right )+240 x^{3} \ln \left (x \right )-72 x^{2} \ln \left (x \right )^{2}-720 x \ln \left (x \right )^{2}-1800 \ln \left (x \right )^{2}}{5 x}\) | \(48\) |
orering | \(\frac {\left (40 x^{12}+500 x^{11}+1124 x^{10}-2590 x^{9}+8480 x^{8}+68025 x^{7}+110691 x^{6}+1979520 x^{5}+2218575 x^{4}-928125 x^{3}+2671875 x^{2}-13950000 x -984375\right ) \left (\left (72 x^{2}-1800\right ) \ln \left (x \right )^{2}+\left (-144 x^{4}-480 x^{3}+144 x^{2}+1440 x +3600\right ) \ln \left (x \right )+40 x^{6}-48 x^{4}-240 x^{3}\right )}{5 x \left (40 x^{12}+100 x^{11}-3076 x^{10}-10830 x^{9}+12280 x^{8}+33225 x^{7}-51309 x^{6}-38640 x^{5}+154275 x^{4}+421875 x^{3}-106875 x^{2}-225000 x +140625\right )}-\frac {x^{2} \left (32 x^{12}+600 x^{11}+3504 x^{10}+4950 x^{9}-11028 x^{8}+1028295 x^{5}+4860900 x^{4}-5211000 x^{3}-10080000 x^{2}+35296875 x +1687500\right ) \left (\frac {144 x \ln \left (x \right )^{2}+\frac {2 \left (72 x^{2}-1800\right ) \ln \left (x \right )}{x}+\left (-576 x^{3}-1440 x^{2}+288 x +1440\right ) \ln \left (x \right )+\frac {-144 x^{4}-480 x^{3}+144 x^{2}+1440 x +3600}{x}+240 x^{5}-192 x^{3}-720 x^{2}}{5 x^{2}}-\frac {2 \left (\left (72 x^{2}-1800\right ) \ln \left (x \right )^{2}+\left (-144 x^{4}-480 x^{3}+144 x^{2}+1440 x +3600\right ) \ln \left (x \right )+40 x^{6}-48 x^{4}-240 x^{3}\right )}{5 x^{3}}\right )}{2 \left (40 x^{12}+100 x^{11}-3076 x^{10}-10830 x^{9}+12280 x^{8}+33225 x^{7}-51309 x^{6}-38640 x^{5}+154275 x^{4}+421875 x^{3}-106875 x^{2}-225000 x +140625\right )}+\frac {\left (16 x^{12}+360 x^{11}+2628 x^{10}+4950 x^{9}-16542 x^{8}+38646 x^{6}-3084885 x^{5}-7291350 x^{4}+5211000 x^{3}+7560000 x^{2}-21178125 x -843750\right ) x^{3} \left (\frac {144 \ln \left (x \right )^{2}+576 \ln \left (x \right )+\frac {144 x^{2}-3600}{x^{2}}-\frac {2 \left (72 x^{2}-1800\right ) \ln \left (x \right )}{x^{2}}+\left (-1728 x^{2}-2880 x +288\right ) \ln \left (x \right )+\frac {-1152 x^{3}-2880 x^{2}+576 x +2880}{x}-\frac {-144 x^{4}-480 x^{3}+144 x^{2}+1440 x +3600}{x^{2}}+1200 x^{4}-576 x^{2}-1440 x}{5 x^{2}}-\frac {4 \left (144 x \ln \left (x \right )^{2}+\frac {2 \left (72 x^{2}-1800\right ) \ln \left (x \right )}{x}+\left (-576 x^{3}-1440 x^{2}+288 x +1440\right ) \ln \left (x \right )+\frac {-144 x^{4}-480 x^{3}+144 x^{2}+1440 x +3600}{x}+240 x^{5}-192 x^{3}-720 x^{2}\right )}{5 x^{3}}+\frac {\frac {6 \left (72 x^{2}-1800\right ) \ln \left (x \right )^{2}}{5}+\frac {6 \left (-144 x^{4}-480 x^{3}+144 x^{2}+1440 x +3600\right ) \ln \left (x \right )}{5}+48 x^{6}-\frac {288 x^{4}}{5}-288 x^{3}}{x^{4}}\right )}{6 \left (20 x^{8}-100 x^{7}-758 x^{6}+420 x^{5}+1103 x^{4}-6150 x^{3}+3750 x -1875\right ) \left (2 x^{4}+15 x^{3}-3 x^{2}-30 x -75\right )}\) | \(811\) |
Input:
int(1/5*((72*x^2-1800)*ln(x)^2+(-144*x^4-480*x^3+144*x^2+1440*x+3600)*ln(x )+40*x^6-48*x^4-240*x^3)/x^2,x,method=_RETURNVERBOSE)
Output:
72/5*(x^2+10*x+25)/x*ln(x)^2+1/5*(-48*x^3-240*x^2)*ln(x)+8/5*x^5
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {-240 x^3-48 x^4+40 x^6+\left (3600+1440 x+144 x^2-480 x^3-144 x^4\right ) \log (x)+\left (-1800+72 x^2\right ) \log ^2(x)}{5 x^2} \, dx=\frac {8 \, {\left (x^{6} + 9 \, {\left (x^{2} + 10 \, x + 25\right )} \log \left (x\right )^{2} - 6 \, {\left (x^{4} + 5 \, x^{3}\right )} \log \left (x\right )\right )}}{5 \, x} \] Input:
integrate(1/5*((72*x^2-1800)*log(x)^2+(-144*x^4-480*x^3+144*x^2+1440*x+360 0)*log(x)+40*x^6-48*x^4-240*x^3)/x^2,x, algorithm="fricas")
Output:
8/5*(x^6 + 9*(x^2 + 10*x + 25)*log(x)^2 - 6*(x^4 + 5*x^3)*log(x))/x
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int \frac {-240 x^3-48 x^4+40 x^6+\left (3600+1440 x+144 x^2-480 x^3-144 x^4\right ) \log (x)+\left (-1800+72 x^2\right ) \log ^2(x)}{5 x^2} \, dx=\frac {8 x^{5}}{5} + \left (- \frac {48 x^{3}}{5} - 48 x^{2}\right ) \log {\left (x \right )} + \frac {\left (72 x^{2} + 720 x + 1800\right ) \log {\left (x \right )}^{2}}{5 x} \] Input:
integrate(1/5*((72*x**2-1800)*ln(x)**2+(-144*x**4-480*x**3+144*x**2+1440*x +3600)*ln(x)+40*x**6-48*x**4-240*x**3)/x**2,x)
Output:
8*x**5/5 + (-48*x**3/5 - 48*x**2)*log(x) + (72*x**2 + 720*x + 1800)*log(x) **2/(5*x)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (19) = 38\).
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22 \[ \int \frac {-240 x^3-48 x^4+40 x^6+\left (3600+1440 x+144 x^2-480 x^3-144 x^4\right ) \log (x)+\left (-1800+72 x^2\right ) \log ^2(x)}{5 x^2} \, dx=\frac {8}{5} \, x^{5} - \frac {48}{5} \, x^{3} \log \left (x\right ) - 48 \, x^{2} \log \left (x\right ) + \frac {72}{5} \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x + \frac {144}{5} \, x \log \left (x\right ) + 144 \, \log \left (x\right )^{2} - \frac {144}{5} \, x + \frac {360 \, {\left (\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 2\right )}}{x} - \frac {720 \, \log \left (x\right )}{x} - \frac {720}{x} \] Input:
integrate(1/5*((72*x^2-1800)*log(x)^2+(-144*x^4-480*x^3+144*x^2+1440*x+360 0)*log(x)+40*x^6-48*x^4-240*x^3)/x^2,x, algorithm="maxima")
Output:
8/5*x^5 - 48/5*x^3*log(x) - 48*x^2*log(x) + 72/5*(log(x)^2 - 2*log(x) + 2) *x + 144/5*x*log(x) + 144*log(x)^2 - 144/5*x + 360*(log(x)^2 + 2*log(x) + 2)/x - 720*log(x)/x - 720/x
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {-240 x^3-48 x^4+40 x^6+\left (3600+1440 x+144 x^2-480 x^3-144 x^4\right ) \log (x)+\left (-1800+72 x^2\right ) \log ^2(x)}{5 x^2} \, dx=\frac {8}{5} \, x^{5} + \frac {72}{5} \, {\left (x + \frac {25}{x} + 10\right )} \log \left (x\right )^{2} - \frac {48}{5} \, {\left (x^{3} + 5 \, x^{2}\right )} \log \left (x\right ) \] Input:
integrate(1/5*((72*x^2-1800)*log(x)^2+(-144*x^4-480*x^3+144*x^2+1440*x+360 0)*log(x)+40*x^6-48*x^4-240*x^3)/x^2,x, algorithm="giac")
Output:
8/5*x^5 + 72/5*(x + 25/x + 10)*log(x)^2 - 48/5*(x^3 + 5*x^2)*log(x)
Time = 2.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-240 x^3-48 x^4+40 x^6+\left (3600+1440 x+144 x^2-480 x^3-144 x^4\right ) \log (x)+\left (-1800+72 x^2\right ) \log ^2(x)}{5 x^2} \, dx=\frac {8\,{\left (15\,\ln \left (x\right )+3\,x\,\ln \left (x\right )-x^3\right )}^2}{5\,x} \] Input:
int(((log(x)*(1440*x + 144*x^2 - 480*x^3 - 144*x^4 + 3600))/5 + (log(x)^2* (72*x^2 - 1800))/5 - 48*x^3 - (48*x^4)/5 + 8*x^6)/x^2,x)
Output:
(8*(15*log(x) + 3*x*log(x) - x^3)^2)/(5*x)
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96 \[ \int \frac {-240 x^3-48 x^4+40 x^6+\left (3600+1440 x+144 x^2-480 x^3-144 x^4\right ) \log (x)+\left (-1800+72 x^2\right ) \log ^2(x)}{5 x^2} \, dx=\frac {\frac {72 \mathrm {log}\left (x \right )^{2} x^{2}}{5}+144 \mathrm {log}\left (x \right )^{2} x +360 \mathrm {log}\left (x \right )^{2}-\frac {48 \,\mathrm {log}\left (x \right ) x^{4}}{5}-48 \,\mathrm {log}\left (x \right ) x^{3}+\frac {8 x^{6}}{5}}{x} \] Input:
int(1/5*((72*x^2-1800)*log(x)^2+(-144*x^4-480*x^3+144*x^2+1440*x+3600)*log (x)+40*x^6-48*x^4-240*x^3)/x^2,x)
Output:
(8*(9*log(x)**2*x**2 + 90*log(x)**2*x + 225*log(x)**2 - 6*log(x)*x**4 - 30 *log(x)*x**3 + x**6))/(5*x)