\(\int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+(25 x^2+80 x^3+114 x^4+80 x^5+25 x^6) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+(25 x^2+80 x^3+114 x^4+80 x^5+25 x^6) \log (x)} \, dx\) [1961]

Optimal result

Integrand size = 110, antiderivative size = 29 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=2+x+\log \left (\frac {6}{x}+\frac {5}{-\frac {2 x}{5}+(1+x)^2}-\log (x)\right ) \] Output:

ln(6/x+5/((1+x)^2-2/5*x)-ln(x))+2+x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=x-\log \left (x \left (5+8 x+5 x^2\right )\right )+\log \left (30+73 x+30 x^2-5 x \log (x)-8 x^2 \log (x)-5 x^3 \log (x)\right ) \] Input:

Integrate[(150 + 355*x + 359*x^2 - 40*x^3 - 375*x^4 - 125*x^5 + (25*x^2 + 
80*x^3 + 114*x^4 + 80*x^5 + 25*x^6)*Log[x])/(-150*x - 605*x^2 - 884*x^3 - 
605*x^4 - 150*x^5 + (25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6)*Log[x]), 
x]
 

Output:

x - Log[x*(5 + 8*x + 5*x^2)] + Log[30 + 73*x + 30*x^2 - 5*x*Log[x] - 8*x^2 
*Log[x] - 5*x^3*Log[x]]
 

Rubi [F]

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.

\(\displaystyle \int \frac {-125 x^5-375 x^4-40 x^3+359 x^2+\left (25 x^6+80 x^5+114 x^4+80 x^3+25 x^2\right ) \log (x)+355 x+150}{-150 x^5-605 x^4-884 x^3-605 x^2+\left (25 x^6+80 x^5+114 x^4+80 x^3+25 x^2\right ) \log (x)-150 x} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {125 x^5+375 x^4+40 x^3-359 x^2-\left (25 x^6+80 x^5+114 x^4+80 x^3+25 x^2\right ) \log (x)-355 x-150}{x \left (5 x^2+8 x+5\right ) \left (-5 x^3 \log (x)+30 x^2-8 x^2 \log (x)+73 x-5 x \log (x)+30\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {25 x^5+230 x^4+844 x^3+964 x^2+505 x+150}{x \left (5 x^2+8 x+5\right ) \left (5 x^3 \log (x)-30 x^2+8 x^2 \log (x)-73 x+5 x \log (x)-30\right )}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 103 \int \frac {1}{5 \log (x) x^3+8 \log (x) x^2-30 x^2+5 \log (x) x-73 x-30}dx-\frac {1250}{3} i \int \frac {1}{(-10 x-(8-6 i)) \left (5 \log (x) x^3+8 \log (x) x^2-30 x^2+5 \log (x) x-73 x-30\right )}dx+30 \int \frac {1}{x \left (5 \log (x) x^3+8 \log (x) x^2-30 x^2+5 \log (x) x-73 x-30\right )}dx+38 \int \frac {x}{5 \log (x) x^3+8 \log (x) x^2-30 x^2+5 \log (x) x-73 x-30}dx+5 \int \frac {x^2}{5 \log (x) x^3+8 \log (x) x^2-30 x^2+5 \log (x) x-73 x-30}dx-\left (200+\frac {800 i}{3}\right ) \int \frac {1}{(10 x+(8-6 i)) \left (5 \log (x) x^3+8 \log (x) x^2-30 x^2+5 \log (x) x-73 x-30\right )}dx-(200+150 i) \int \frac {1}{(10 x+(8+6 i)) \left (5 \log (x) x^3+8 \log (x) x^2-30 x^2+5 \log (x) x-73 x-30\right )}dx+x\)

Input:

Int[(150 + 355*x + 359*x^2 - 40*x^3 - 375*x^4 - 125*x^5 + (25*x^2 + 80*x^3 
 + 114*x^4 + 80*x^5 + 25*x^6)*Log[x])/(-150*x - 605*x^2 - 884*x^3 - 605*x^ 
4 - 150*x^5 + (25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6)*Log[x]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17

method	result	size
risch	\(x +\ln \left (\ln \left (x \right )-\frac {30 x^{2}+73 x +30}{x \left (5 x^{2}+8 x +5\right )}\right )\)	\(34\)
parallelrisch	\(-\ln \left (x^{2}+\frac {8}{5} x +1\right )+\ln \left (x^{3} \ln \left (x \right )+\frac {8 x^{2} \ln \left (x \right )}{5}-6 x^{2}+x \ln \left (x \right )-\frac {73 x}{5}-6\right )+x -\ln \left (x \right )\)	\(46\)
default	\(-\ln \left (x \right )+x -\ln \left (5 x^{2}+8 x +5\right )+\ln \left (5 x^{3} \ln \left (x \right )+8 x^{2} \ln \left (x \right )+5 x \ln \left (x \right )-30 x^{2}-73 x -30\right )\)	\(50\)
norman	\(-\ln \left (x \right )+x -\ln \left (5 x^{2}+8 x +5\right )+\ln \left (5 x^{3} \ln \left (x \right )+8 x^{2} \ln \left (x \right )+5 x \ln \left (x \right )-30 x^{2}-73 x -30\right )\)	\(50\)

Input:

int(((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*ln(x)-125*x^5-375*x^4-40*x^3+35 
9*x^2+355*x+150)/((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*ln(x)-150*x^5-605* 
x^4-884*x^3-605*x^2-150*x),x,method=_RETURNVERBOSE)
 

Output:

x+ln(ln(x)-(30*x^2+73*x+30)/x/(5*x^2+8*x+5))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=x + \log \left (-\frac {30 \, x^{2} - {\left (5 \, x^{3} + 8 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 73 \, x + 30}{5 \, x^{3} + 8 \, x^{2} + 5 \, x}\right ) \] Input:

integrate(((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-125*x^5-375*x^4-40 
*x^3+359*x^2+355*x+150)/((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-150* 
x^5-605*x^4-884*x^3-605*x^2-150*x),x, algorithm="fricas")
 

Output:

x + log(-(30*x^2 - (5*x^3 + 8*x^2 + 5*x)*log(x) + 73*x + 30)/(5*x^3 + 8*x^ 
2 + 5*x))
 

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=x + \log {\left (\frac {- 30 x^{2} - 73 x - 30}{5 x^{3} + 8 x^{2} + 5 x} + \log {\left (x \right )} \right )} \] Input:

integrate(((25*x**6+80*x**5+114*x**4+80*x**3+25*x**2)*ln(x)-125*x**5-375*x 
**4-40*x**3+359*x**2+355*x+150)/((25*x**6+80*x**5+114*x**4+80*x**3+25*x**2 
)*ln(x)-150*x**5-605*x**4-884*x**3-605*x**2-150*x),x)
 

Output:

x + log((-30*x**2 - 73*x - 30)/(5*x**3 + 8*x**2 + 5*x) + log(x))
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=x + \log \left (-\frac {30 \, x^{2} - {\left (5 \, x^{3} + 8 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 73 \, x + 30}{5 \, x^{3} + 8 \, x^{2} + 5 \, x}\right ) \] Input:

integrate(((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-125*x^5-375*x^4-40 
*x^3+359*x^2+355*x+150)/((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-150* 
x^5-605*x^4-884*x^3-605*x^2-150*x),x, algorithm="maxima")
 

Output:

x + log(-(30*x^2 - (5*x^3 + 8*x^2 + 5*x)*log(x) + 73*x + 30)/(5*x^3 + 8*x^ 
2 + 5*x))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=x + \log \left (5 \, x^{3} \log \left (x\right ) + 8 \, x^{2} \log \left (x\right ) - 30 \, x^{2} + 5 \, x \log \left (x\right ) - 73 \, x - 30\right ) - \log \left (5 \, x^{2} + 8 \, x + 5\right ) - \log \left (x\right ) \] Input:

integrate(((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-125*x^5-375*x^4-40 
*x^3+359*x^2+355*x+150)/((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-150* 
x^5-605*x^4-884*x^3-605*x^2-150*x),x, algorithm="giac")
 

Output:

x + log(5*x^3*log(x) + 8*x^2*log(x) - 30*x^2 + 5*x*log(x) - 73*x - 30) - l 
og(5*x^2 + 8*x + 5) - log(x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=\int -\frac {355\,x+\ln \left (x\right )\,\left (25\,x^6+80\,x^5+114\,x^4+80\,x^3+25\,x^2\right )+359\,x^2-40\,x^3-375\,x^4-125\,x^5+150}{150\,x-\ln \left (x\right )\,\left (25\,x^6+80\,x^5+114\,x^4+80\,x^3+25\,x^2\right )+605\,x^2+884\,x^3+605\,x^4+150\,x^5} \,d x \] Input:

int(-(355*x + log(x)*(25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6) + 359*x 
^2 - 40*x^3 - 375*x^4 - 125*x^5 + 150)/(150*x - log(x)*(25*x^2 + 80*x^3 + 
114*x^4 + 80*x^5 + 25*x^6) + 605*x^2 + 884*x^3 + 605*x^4 + 150*x^5),x)
 

Output:

int(-(355*x + log(x)*(25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6) + 359*x 
^2 - 40*x^3 - 375*x^4 - 125*x^5 + 150)/(150*x - log(x)*(25*x^2 + 80*x^3 + 
114*x^4 + 80*x^5 + 25*x^6) + 605*x^2 + 884*x^3 + 605*x^4 + 150*x^5), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=-\mathrm {log}\left (5 x^{2}+8 x +5\right )+\mathrm {log}\left (5 \,\mathrm {log}\left (x \right ) x^{3}+8 \,\mathrm {log}\left (x \right ) x^{2}+5 \,\mathrm {log}\left (x \right ) x -30 x^{2}-73 x -30\right )-\mathrm {log}\left (x \right )+x \] Input:

int(((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-125*x^5-375*x^4-40*x^3+3 
59*x^2+355*x+150)/((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-150*x^5-60 
5*x^4-884*x^3-605*x^2-150*x),x)
 

Output:

 - log(5*x**2 + 8*x + 5) + log(5*log(x)*x**3 + 8*log(x)*x**2 + 5*log(x)*x 
- 30*x**2 - 73*x - 30) - log(x) + x