Integrand size = 91, antiderivative size = 30 \[ \int \frac {248+398 x+216 x^2+24 x^3+\left (2120+830 x+180 x^2\right ) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+\left (540+360 x+60 x^2\right ) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(x)} \, dx=\frac {x+x^2 \left (3+\frac {4-x}{\left (\frac {2 (3+x)}{5}+\log (x)\right )^2}\right )}{x} \] Output:
(x+x^2*(3+(4-x)/(6/5+2/5*x+ln(x))^2))/x
Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \frac {248+398 x+216 x^2+24 x^3+\left (2120+830 x+180 x^2\right ) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+\left (540+360 x+60 x^2\right ) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(x)} \, dx=3 x-\frac {25 (-4+x) x}{(6+2 x+5 \log (x))^2} \] Input:
Integrate[(248 + 398*x + 216*x^2 + 24*x^3 + (2120 + 830*x + 180*x^2)*Log[x ] + (1350 + 450*x)*Log[x]^2 + 375*Log[x]^3)/(216 + 216*x + 72*x^2 + 8*x^3 + (540 + 360*x + 60*x^2)*Log[x] + (450 + 150*x)*Log[x]^2 + 125*Log[x]^3),x ]
Output:
3*x - (25*(-4 + x)*x)/(6 + 2*x + 5*Log[x])^2
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.
\(\displaystyle \int \frac {24 x^3+216 x^2+\left (180 x^2+830 x+2120\right ) \log (x)+398 x+375 \log ^3(x)+(450 x+1350) \log ^2(x)+248}{8 x^3+72 x^2+\left (60 x^2+360 x+540\right ) \log (x)+216 x+125 \log ^3(x)+(150 x+450) \log ^2(x)+216} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {24 x^3+216 x^2+\left (180 x^2+830 x+2120\right ) \log (x)+398 x+375 \log ^3(x)+(450 x+1350) \log ^2(x)+248}{(2 x+5 \log (x)+6)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {50 \left (2 x^2-3 x-20\right )}{(2 x+5 \log (x)+6)^3}-\frac {50 (x-2)}{(2 x+5 \log (x)+6)^2}+3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 100 \int \frac {x^2}{(2 x+5 \log (x)+6)^3}dx-1000 \int \frac {1}{(2 x+5 \log (x)+6)^3}dx-150 \int \frac {x}{(2 x+5 \log (x)+6)^3}dx+100 \int \frac {1}{(2 x+5 \log (x)+6)^2}dx-50 \int \frac {x}{(2 x+5 \log (x)+6)^2}dx+3 x\) |
Input:
Int[(248 + 398*x + 216*x^2 + 24*x^3 + (2120 + 830*x + 180*x^2)*Log[x] + (1 350 + 450*x)*Log[x]^2 + 375*Log[x]^3)/(216 + 216*x + 72*x^2 + 8*x^3 + (540 + 360*x + 60*x^2)*Log[x] + (450 + 150*x)*Log[x]^2 + 125*Log[x]^3),x]
Output:
$Aborted
Time = 1.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
risch | \(3 x -\frac {25 \left (x -4\right ) x}{\left (5 \ln \left (x \right )+2 x +6\right )^{2}}\) | \(22\) |
default | \(\frac {47 x^{2}+208 x +180 x \ln \left (x \right )+12 x^{3}+75 x \ln \left (x \right )^{2}+60 x^{2} \ln \left (x \right )}{\left (5 \ln \left (x \right )+2 x +6\right )^{2}}\) | \(46\) |
norman | \(\frac {-705 \ln \left (x \right )-74 x -\frac {1175 \ln \left (x \right )^{2}}{4}-55 x \ln \left (x \right )+12 x^{3}+75 x \ln \left (x \right )^{2}+60 x^{2} \ln \left (x \right )-423}{\left (5 \ln \left (x \right )+2 x +6\right )^{2}}\) | \(52\) |
parallelrisch | \(\frac {5200 x +4500 x \ln \left (x \right )+1875 x \ln \left (x \right )^{2}+1500 x^{2} \ln \left (x \right )+1175 x^{2}+300 x^{3}}{100 x^{2}+500 x \ln \left (x \right )+625 \ln \left (x \right )^{2}+600 x +1500 \ln \left (x \right )+900}\) | \(63\) |
Input:
int((375*ln(x)^3+(450*x+1350)*ln(x)^2+(180*x^2+830*x+2120)*ln(x)+24*x^3+21 6*x^2+398*x+248)/(125*ln(x)^3+(150*x+450)*ln(x)^2+(60*x^2+360*x+540)*ln(x) +8*x^3+72*x^2+216*x+216),x,method=_RETURNVERBOSE)
Output:
3*x-25*(x-4)*x/(5*ln(x)+2*x+6)^2
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {248+398 x+216 x^2+24 x^3+\left (2120+830 x+180 x^2\right ) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+\left (540+360 x+60 x^2\right ) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(x)} \, dx=\frac {12 \, x^{3} + 75 \, x \log \left (x\right )^{2} + 47 \, x^{2} + 60 \, {\left (x^{2} + 3 \, x\right )} \log \left (x\right ) + 208 \, x}{4 \, x^{2} + 20 \, {\left (x + 3\right )} \log \left (x\right ) + 25 \, \log \left (x\right )^{2} + 24 \, x + 36} \] Input:
integrate((375*log(x)^3+(450*x+1350)*log(x)^2+(180*x^2+830*x+2120)*log(x)+ 24*x^3+216*x^2+398*x+248)/(125*log(x)^3+(150*x+450)*log(x)^2+(60*x^2+360*x +540)*log(x)+8*x^3+72*x^2+216*x+216),x, algorithm="fricas")
Output:
(12*x^3 + 75*x*log(x)^2 + 47*x^2 + 60*(x^2 + 3*x)*log(x) + 208*x)/(4*x^2 + 20*(x + 3)*log(x) + 25*log(x)^2 + 24*x + 36)
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {248+398 x+216 x^2+24 x^3+\left (2120+830 x+180 x^2\right ) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+\left (540+360 x+60 x^2\right ) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(x)} \, dx=3 x + \frac {- x^{2} + 4 x}{\frac {4 x^{2}}{25} + \frac {24 x}{25} + \left (\frac {4 x}{5} + \frac {12}{5}\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + \frac {36}{25}} \] Input:
integrate((375*ln(x)**3+(450*x+1350)*ln(x)**2+(180*x**2+830*x+2120)*ln(x)+ 24*x**3+216*x**2+398*x+248)/(125*ln(x)**3+(150*x+450)*ln(x)**2+(60*x**2+36 0*x+540)*ln(x)+8*x**3+72*x**2+216*x+216),x)
Output:
3*x + (-x**2 + 4*x)/(4*x**2/25 + 24*x/25 + (4*x/5 + 12/5)*log(x) + log(x)* *2 + 36/25)
Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {248+398 x+216 x^2+24 x^3+\left (2120+830 x+180 x^2\right ) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+\left (540+360 x+60 x^2\right ) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(x)} \, dx=\frac {12 \, x^{3} + 75 \, x \log \left (x\right )^{2} + 47 \, x^{2} + 60 \, {\left (x^{2} + 3 \, x\right )} \log \left (x\right ) + 208 \, x}{4 \, x^{2} + 20 \, {\left (x + 3\right )} \log \left (x\right ) + 25 \, \log \left (x\right )^{2} + 24 \, x + 36} \] Input:
integrate((375*log(x)^3+(450*x+1350)*log(x)^2+(180*x^2+830*x+2120)*log(x)+ 24*x^3+216*x^2+398*x+248)/(125*log(x)^3+(150*x+450)*log(x)^2+(60*x^2+360*x +540)*log(x)+8*x^3+72*x^2+216*x+216),x, algorithm="maxima")
Output:
(12*x^3 + 75*x*log(x)^2 + 47*x^2 + 60*(x^2 + 3*x)*log(x) + 208*x)/(4*x^2 + 20*(x + 3)*log(x) + 25*log(x)^2 + 24*x + 36)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (31) = 62\).
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \frac {248+398 x+216 x^2+24 x^3+\left (2120+830 x+180 x^2\right ) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+\left (540+360 x+60 x^2\right ) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(x)} \, dx=3 \, x - \frac {25 \, {\left (2 \, x^{3} - 3 \, x^{2} - 20 \, x\right )}}{8 \, x^{3} + 40 \, x^{2} \log \left (x\right ) + 50 \, x \log \left (x\right )^{2} + 68 \, x^{2} + 220 \, x \log \left (x\right ) + 125 \, \log \left (x\right )^{2} + 192 \, x + 300 \, \log \left (x\right ) + 180} \] Input:
integrate((375*log(x)^3+(450*x+1350)*log(x)^2+(180*x^2+830*x+2120)*log(x)+ 24*x^3+216*x^2+398*x+248)/(125*log(x)^3+(150*x+450)*log(x)^2+(60*x^2+360*x +540)*log(x)+8*x^3+72*x^2+216*x+216),x, algorithm="giac")
Output:
3*x - 25*(2*x^3 - 3*x^2 - 20*x)/(8*x^3 + 40*x^2*log(x) + 50*x*log(x)^2 + 6 8*x^2 + 220*x*log(x) + 125*log(x)^2 + 192*x + 300*log(x) + 180)
Time = 4.46 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {248+398 x+216 x^2+24 x^3+\left (2120+830 x+180 x^2\right ) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+\left (540+360 x+60 x^2\right ) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(x)} \, dx=\frac {x\,\left (12\,x^2+60\,x\,\ln \left (x\right )+47\,x+75\,{\ln \left (x\right )}^2+180\,\ln \left (x\right )+208\right )}{{\left (2\,x+5\,\ln \left (x\right )+6\right )}^2} \] Input:
int((398*x + 375*log(x)^3 + log(x)*(830*x + 180*x^2 + 2120) + 216*x^2 + 24 *x^3 + log(x)^2*(450*x + 1350) + 248)/(216*x + 125*log(x)^3 + log(x)*(360* x + 60*x^2 + 540) + 72*x^2 + 8*x^3 + log(x)^2*(150*x + 450) + 216),x)
Output:
(x*(47*x + 180*log(x) + 75*log(x)^2 + 60*x*log(x) + 12*x^2 + 208))/(2*x + 5*log(x) + 6)^2
Time = 0.17 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.30 \[ \int \frac {248+398 x+216 x^2+24 x^3+\left (2120+830 x+180 x^2\right ) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+\left (540+360 x+60 x^2\right ) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(x)} \, dx=\frac {225 \mathrm {log}\left (x \right )^{2} x -650 \mathrm {log}\left (x \right )^{2}+180 \,\mathrm {log}\left (x \right ) x^{2}+20 \,\mathrm {log}\left (x \right ) x -1560 \,\mathrm {log}\left (x \right )+36 x^{3}+37 x^{2}-936}{75 \mathrm {log}\left (x \right )^{2}+60 \,\mathrm {log}\left (x \right ) x +180 \,\mathrm {log}\left (x \right )+12 x^{2}+72 x +108} \] Input:
int((375*log(x)^3+(450*x+1350)*log(x)^2+(180*x^2+830*x+2120)*log(x)+24*x^3 +216*x^2+398*x+248)/(125*log(x)^3+(150*x+450)*log(x)^2+(60*x^2+360*x+540)* log(x)+8*x^3+72*x^2+216*x+216),x)
Output:
(225*log(x)**2*x - 650*log(x)**2 + 180*log(x)*x**2 + 20*log(x)*x - 1560*lo g(x) + 36*x**3 + 37*x**2 - 936)/(3*(25*log(x)**2 + 20*log(x)*x + 60*log(x) + 4*x**2 + 24*x + 36))