Integrand size = 147, antiderivative size = 32 \[ \int \frac {512+1920 x+2528 x^2+1480 x^3+614 x^4+255 x^5+\left (-384-960 x-696 x^2-240 x^3-153 x^4\right ) \log (x)+\left (96+120 x+24 x^2+30 x^3\right ) \log ^2(x)+\left (-8-2 x^2\right ) \log ^3(x)}{-64 x^2-240 x^3-300 x^4-125 x^5+\left (48 x^2+120 x^3+75 x^4\right ) \log (x)+\left (-12 x^2-15 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=4+2 \left (-1+\frac {4}{x}\right )-2 x-\frac {x}{\left (-5-\frac {4}{x}+\frac {\log (x)}{x}\right )^2} \]
Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {512+1920 x+2528 x^2+1480 x^3+614 x^4+255 x^5+\left (-384-960 x-696 x^2-240 x^3-153 x^4\right ) \log (x)+\left (96+120 x+24 x^2+30 x^3\right ) \log ^2(x)+\left (-8-2 x^2\right ) \log ^3(x)}{-64 x^2-240 x^3-300 x^4-125 x^5+\left (48 x^2+120 x^3+75 x^4\right ) \log (x)+\left (-12 x^2-15 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {8}{x}-2 x-\frac {x^3}{(-4-5 x+\log (x))^2} \]
Integrate[(512 + 1920*x + 2528*x^2 + 1480*x^3 + 614*x^4 + 255*x^5 + (-384 - 960*x - 696*x^2 - 240*x^3 - 153*x^4)*Log[x] + (96 + 120*x + 24*x^2 + 30* x^3)*Log[x]^2 + (-8 - 2*x^2)*Log[x]^3)/(-64*x^2 - 240*x^3 - 300*x^4 - 125* x^5 + (48*x^2 + 120*x^3 + 75*x^4)*Log[x] + (-12*x^2 - 15*x^3)*Log[x]^2 + x ^2*Log[x]^3),x]
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.
\(\displaystyle \int \frac {255 x^5+614 x^4+1480 x^3+2528 x^2+\left (-2 x^2-8\right ) \log ^3(x)+\left (30 x^3+24 x^2+120 x+96\right ) \log ^2(x)+\left (-153 x^4-240 x^3-696 x^2-960 x-384\right ) \log (x)+1920 x+512}{-125 x^5-300 x^4-240 x^3-64 x^2+x^2 \log ^3(x)+\left (-15 x^3-12 x^2\right ) \log ^2(x)+\left (75 x^4+120 x^3+48 x^2\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-255 x^5-614 x^4-1480 x^3-2528 x^2-\left (-2 x^2-8\right ) \log ^3(x)-\left (30 x^3+24 x^2+120 x+96\right ) \log ^2(x)-\left (-153 x^4-240 x^3-696 x^2-960 x-384\right ) \log (x)-1920 x-512}{x^2 (5 x-\log (x)+4)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 \left (x^2+4\right )}{x^2}-\frac {3 x^2}{(5 x-\log (x)+4)^2}+\frac {2 (5 x-1) x^2}{(5 x-\log (x)+4)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 10 \int \frac {x^3}{(5 x-\log (x)+4)^3}dx-2 \int \frac {x^2}{(5 x-\log (x)+4)^3}dx-3 \int \frac {x^2}{(5 x-\log (x)+4)^2}dx-2 x+\frac {8}{x}\) |
Int[(512 + 1920*x + 2528*x^2 + 1480*x^3 + 614*x^4 + 255*x^5 + (-384 - 960* x - 696*x^2 - 240*x^3 - 153*x^4)*Log[x] + (96 + 120*x + 24*x^2 + 30*x^3)*L og[x]^2 + (-8 - 2*x^2)*Log[x]^3)/(-64*x^2 - 240*x^3 - 300*x^4 - 125*x^5 + (48*x^2 + 120*x^3 + 75*x^4)*Log[x] + (-12*x^2 - 15*x^3)*Log[x]^2 + x^2*Log [x]^3),x]
3.27.74.3.1 Defintions of rubi rules used
Time = 0.59 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88
method | result | size |
risch | \(-\frac {2 \left (x^{2}-4\right )}{x}-\frac {x^{3}}{\left (5 x -\ln \left (x \right )+4\right )^{2}}\) | \(28\) |
default | \(-\frac {-128-320 x -168 x^{2}+80 x^{3}-16 x^{2} \ln \left (x \right )+80 x \ln \left (x \right )+64 \ln \left (x \right )-8 \ln \left (x \right )^{2}+51 x^{4}+2 x^{2} \ln \left (x \right )^{2}-20 x^{3} \ln \left (x \right )}{x \left (\ln \left (x \right )-5 x -4\right )^{2}}\) | \(73\) |
norman | \(\frac {128+296 x^{2}+\frac {1856 x}{5}-2 x^{2} \ln \left (x \right )^{2}+20 x^{3} \ln \left (x \right )-16 x^{2} \ln \left (x \right )-\frac {528 x \ln \left (x \right )}{5}+\frac {16 x \ln \left (x \right )^{2}}{5}-51 x^{4}+8 \ln \left (x \right )^{2}-64 \ln \left (x \right )}{x \left (5 x -\ln \left (x \right )+4\right )^{2}}\) | \(76\) |
parallelrisch | \(\frac {128+320 x -80 x \ln \left (x \right )-2 x^{2} \ln \left (x \right )^{2}-64 \ln \left (x \right )+8 \ln \left (x \right )^{2}-51 x^{4}-80 x^{3}+168 x^{2}+20 x^{3} \ln \left (x \right )+16 x^{2} \ln \left (x \right )}{x \left (25 x^{2}-10 x \ln \left (x \right )+\ln \left (x \right )^{2}+40 x -8 \ln \left (x \right )+16\right )}\) | \(88\) |
int(((-2*x^2-8)*ln(x)^3+(30*x^3+24*x^2+120*x+96)*ln(x)^2+(-153*x^4-240*x^3 -696*x^2-960*x-384)*ln(x)+255*x^5+614*x^4+1480*x^3+2528*x^2+1920*x+512)/(x ^2*ln(x)^3+(-15*x^3-12*x^2)*ln(x)^2+(75*x^4+120*x^3+48*x^2)*ln(x)-125*x^5- 300*x^4-240*x^3-64*x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.72 \[ \int \frac {512+1920 x+2528 x^2+1480 x^3+614 x^4+255 x^5+\left (-384-960 x-696 x^2-240 x^3-153 x^4\right ) \log (x)+\left (96+120 x+24 x^2+30 x^3\right ) \log ^2(x)+\left (-8-2 x^2\right ) \log ^3(x)}{-64 x^2-240 x^3-300 x^4-125 x^5+\left (48 x^2+120 x^3+75 x^4\right ) \log (x)+\left (-12 x^2-15 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=-\frac {51 \, x^{4} + 80 \, x^{3} + 2 \, {\left (x^{2} - 4\right )} \log \left (x\right )^{2} - 168 \, x^{2} - 4 \, {\left (5 \, x^{3} + 4 \, x^{2} - 20 \, x - 16\right )} \log \left (x\right ) - 320 \, x - 128}{25 \, x^{3} + x \log \left (x\right )^{2} + 40 \, x^{2} - 2 \, {\left (5 \, x^{2} + 4 \, x\right )} \log \left (x\right ) + 16 \, x} \]
integrate(((-2*x^2-8)*log(x)^3+(30*x^3+24*x^2+120*x+96)*log(x)^2+(-153*x^4 -240*x^3-696*x^2-960*x-384)*log(x)+255*x^5+614*x^4+1480*x^3+2528*x^2+1920* x+512)/(x^2*log(x)^3+(-15*x^3-12*x^2)*log(x)^2+(75*x^4+120*x^3+48*x^2)*log (x)-125*x^5-300*x^4-240*x^3-64*x^2),x, algorithm=\
-(51*x^4 + 80*x^3 + 2*(x^2 - 4)*log(x)^2 - 168*x^2 - 4*(5*x^3 + 4*x^2 - 20 *x - 16)*log(x) - 320*x - 128)/(25*x^3 + x*log(x)^2 + 40*x^2 - 2*(5*x^2 + 4*x)*log(x) + 16*x)
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {512+1920 x+2528 x^2+1480 x^3+614 x^4+255 x^5+\left (-384-960 x-696 x^2-240 x^3-153 x^4\right ) \log (x)+\left (96+120 x+24 x^2+30 x^3\right ) \log ^2(x)+\left (-8-2 x^2\right ) \log ^3(x)}{-64 x^2-240 x^3-300 x^4-125 x^5+\left (48 x^2+120 x^3+75 x^4\right ) \log (x)+\left (-12 x^2-15 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=- \frac {x^{3}}{25 x^{2} + 40 x + \left (- 10 x - 8\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + 16} - 2 x + \frac {8}{x} \]
integrate(((-2*x**2-8)*ln(x)**3+(30*x**3+24*x**2+120*x+96)*ln(x)**2+(-153* x**4-240*x**3-696*x**2-960*x-384)*ln(x)+255*x**5+614*x**4+1480*x**3+2528*x **2+1920*x+512)/(x**2*ln(x)**3+(-15*x**3-12*x**2)*ln(x)**2+(75*x**4+120*x* *3+48*x**2)*ln(x)-125*x**5-300*x**4-240*x**3-64*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (28) = 56\).
Time = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.72 \[ \int \frac {512+1920 x+2528 x^2+1480 x^3+614 x^4+255 x^5+\left (-384-960 x-696 x^2-240 x^3-153 x^4\right ) \log (x)+\left (96+120 x+24 x^2+30 x^3\right ) \log ^2(x)+\left (-8-2 x^2\right ) \log ^3(x)}{-64 x^2-240 x^3-300 x^4-125 x^5+\left (48 x^2+120 x^3+75 x^4\right ) \log (x)+\left (-12 x^2-15 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=-\frac {51 \, x^{4} + 80 \, x^{3} + 2 \, {\left (x^{2} - 4\right )} \log \left (x\right )^{2} - 168 \, x^{2} - 4 \, {\left (5 \, x^{3} + 4 \, x^{2} - 20 \, x - 16\right )} \log \left (x\right ) - 320 \, x - 128}{25 \, x^{3} + x \log \left (x\right )^{2} + 40 \, x^{2} - 2 \, {\left (5 \, x^{2} + 4 \, x\right )} \log \left (x\right ) + 16 \, x} \]
integrate(((-2*x^2-8)*log(x)^3+(30*x^3+24*x^2+120*x+96)*log(x)^2+(-153*x^4 -240*x^3-696*x^2-960*x-384)*log(x)+255*x^5+614*x^4+1480*x^3+2528*x^2+1920* x+512)/(x^2*log(x)^3+(-15*x^3-12*x^2)*log(x)^2+(75*x^4+120*x^3+48*x^2)*log (x)-125*x^5-300*x^4-240*x^3-64*x^2),x, algorithm=\
-(51*x^4 + 80*x^3 + 2*(x^2 - 4)*log(x)^2 - 168*x^2 - 4*(5*x^3 + 4*x^2 - 20 *x - 16)*log(x) - 320*x - 128)/(25*x^3 + x*log(x)^2 + 40*x^2 - 2*(5*x^2 + 4*x)*log(x) + 16*x)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {512+1920 x+2528 x^2+1480 x^3+614 x^4+255 x^5+\left (-384-960 x-696 x^2-240 x^3-153 x^4\right ) \log (x)+\left (96+120 x+24 x^2+30 x^3\right ) \log ^2(x)+\left (-8-2 x^2\right ) \log ^3(x)}{-64 x^2-240 x^3-300 x^4-125 x^5+\left (48 x^2+120 x^3+75 x^4\right ) \log (x)+\left (-12 x^2-15 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=-2 \, x - \frac {5 \, x^{4} - x^{3}}{125 \, x^{3} - 50 \, x^{2} \log \left (x\right ) + 5 \, x \log \left (x\right )^{2} + 175 \, x^{2} - 30 \, x \log \left (x\right ) - \log \left (x\right )^{2} + 40 \, x + 8 \, \log \left (x\right ) - 16} + \frac {8}{x} \]
integrate(((-2*x^2-8)*log(x)^3+(30*x^3+24*x^2+120*x+96)*log(x)^2+(-153*x^4 -240*x^3-696*x^2-960*x-384)*log(x)+255*x^5+614*x^4+1480*x^3+2528*x^2+1920* x+512)/(x^2*log(x)^3+(-15*x^3-12*x^2)*log(x)^2+(75*x^4+120*x^3+48*x^2)*log (x)-125*x^5-300*x^4-240*x^3-64*x^2),x, algorithm=\
-2*x - (5*x^4 - x^3)/(125*x^3 - 50*x^2*log(x) + 5*x*log(x)^2 + 175*x^2 - 3 0*x*log(x) - log(x)^2 + 40*x + 8*log(x) - 16) + 8/x
Timed out. \[ \int \frac {512+1920 x+2528 x^2+1480 x^3+614 x^4+255 x^5+\left (-384-960 x-696 x^2-240 x^3-153 x^4\right ) \log (x)+\left (96+120 x+24 x^2+30 x^3\right ) \log ^2(x)+\left (-8-2 x^2\right ) \log ^3(x)}{-64 x^2-240 x^3-300 x^4-125 x^5+\left (48 x^2+120 x^3+75 x^4\right ) \log (x)+\left (-12 x^2-15 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\int -\frac {1920\,x-\ln \left (x\right )\,\left (153\,x^4+240\,x^3+696\,x^2+960\,x+384\right )-{\ln \left (x\right )}^3\,\left (2\,x^2+8\right )+{\ln \left (x\right )}^2\,\left (30\,x^3+24\,x^2+120\,x+96\right )+2528\,x^2+1480\,x^3+614\,x^4+255\,x^5+512}{{\ln \left (x\right )}^2\,\left (15\,x^3+12\,x^2\right )-\ln \left (x\right )\,\left (75\,x^4+120\,x^3+48\,x^2\right )-x^2\,{\ln \left (x\right )}^3+64\,x^2+240\,x^3+300\,x^4+125\,x^5} \,d x \]
int(-(1920*x - log(x)*(960*x + 696*x^2 + 240*x^3 + 153*x^4 + 384) - log(x) ^3*(2*x^2 + 8) + log(x)^2*(120*x + 24*x^2 + 30*x^3 + 96) + 2528*x^2 + 1480 *x^3 + 614*x^4 + 255*x^5 + 512)/(log(x)^2*(12*x^2 + 15*x^3) - log(x)*(48*x ^2 + 120*x^3 + 75*x^4) - x^2*log(x)^3 + 64*x^2 + 240*x^3 + 300*x^4 + 125*x ^5),x)
int(-(1920*x - log(x)*(960*x + 696*x^2 + 240*x^3 + 153*x^4 + 384) - log(x) ^3*(2*x^2 + 8) + log(x)^2*(120*x + 24*x^2 + 30*x^3 + 96) + 2528*x^2 + 1480 *x^3 + 614*x^4 + 255*x^5 + 512)/(log(x)^2*(12*x^2 + 15*x^3) - log(x)*(48*x ^2 + 120*x^3 + 75*x^4) - x^2*log(x)^3 + 64*x^2 + 240*x^3 + 300*x^4 + 125*x ^5), x)