\(\int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+(2 x-500 x^2+100 x^3+62500 x^6) \log (x)+(-10+3750 x^4) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+(-500 x^2+100 x^3+62500 x^6) \log (x)+(-10+2 x+3750 x^4) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx\) [1281]

Optimal result

Integrand size = 142, antiderivative size = 22 \[ \int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+\left (2 x-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+\left (-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+2 x+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx=\frac {x}{1+\frac {x}{-5+\left (25 x^2+\log (x)\right )^2}} \] Output:

x/(x/((ln(x)+25*x^2)^2-5)+1)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+\left (2 x-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+\left (-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+2 x+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx=x-\frac {x^2}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)} \] Input:

Integrate[(25 + 50*x^3 - 6250*x^4 + 2500*x^5 + 390625*x^8 + (2*x - 500*x^2 
 + 100*x^3 + 62500*x^6)*Log[x] + (-10 + 3750*x^4)*Log[x]^2 + 100*x^2*Log[x 
]^3 + Log[x]^4)/(25 - 10*x + x^2 - 6250*x^4 + 1250*x^5 + 390625*x^8 + (-50 
0*x^2 + 100*x^3 + 62500*x^6)*Log[x] + (-10 + 2*x + 3750*x^4)*Log[x]^2 + 10 
0*x^2*Log[x]^3 + Log[x]^4),x]
 

Output:

x - x^2/(-5 + x + 625*x^4 + 50*x^2*Log[x] + Log[x]^2)
 

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.

\(\displaystyle \int \frac {390625 x^8+2500 x^5-6250 x^4+\left (3750 x^4-10\right ) \log ^2(x)+50 x^3+100 x^2 \log ^3(x)+\left (62500 x^6+100 x^3-500 x^2+2 x\right ) \log (x)+\log ^4(x)+25}{390625 x^8+1250 x^5-6250 x^4+\left (3750 x^4+2 x-10\right ) \log ^2(x)+x^2+100 x^2 \log ^3(x)+\left (62500 x^6+100 x^3-500 x^2\right ) \log (x)-10 x+\log ^4(x)+25} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {390625 x^8+2500 x^5-6250 x^4+\left (3750 x^4-10\right ) \log ^2(x)+50 x^3+100 x^2 \log ^3(x)+\left (62500 x^6+100 x^3-500 x^2+2 x\right ) \log (x)+\log ^4(x)+25}{\left (-625 x^4-50 x^2 \log (x)-x-\log ^2(x)+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {2 x}{625 x^4+50 x^2 \log (x)+x+\log ^2(x)-5}+\frac {x \left (2500 x^4+50 x^2+100 x^2 \log (x)+x+2 \log (x)\right )}{\left (625 x^4+50 x^2 \log (x)+x+\log ^2(x)-5\right )^2}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {x^2}{\left (625 x^4+50 \log (x) x^2+x+\log ^2(x)-5\right )^2}dx+2 \int \frac {x \log (x)}{\left (625 x^4+50 \log (x) x^2+x+\log ^2(x)-5\right )^2}dx-2 \int \frac {x}{625 x^4+50 \log (x) x^2+x+\log ^2(x)-5}dx+2500 \int \frac {x^5}{\left (625 x^4+50 \log (x) x^2+x+\log ^2(x)-5\right )^2}dx+50 \int \frac {x^3}{\left (625 x^4+50 \log (x) x^2+x+\log ^2(x)-5\right )^2}dx+100 \int \frac {x^3 \log (x)}{\left (625 x^4+50 \log (x) x^2+x+\log ^2(x)-5\right )^2}dx+x\)

Input:

Int[(25 + 50*x^3 - 6250*x^4 + 2500*x^5 + 390625*x^8 + (2*x - 500*x^2 + 100 
*x^3 + 62500*x^6)*Log[x] + (-10 + 3750*x^4)*Log[x]^2 + 100*x^2*Log[x]^3 + 
Log[x]^4)/(25 - 10*x + x^2 - 6250*x^4 + 1250*x^5 + 390625*x^8 + (-500*x^2 
+ 100*x^3 + 62500*x^6)*Log[x] + (-10 + 2*x + 3750*x^4)*Log[x]^2 + 100*x^2* 
Log[x]^3 + Log[x]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 11.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32

method	result	size
default	\(x -\frac {x^{2}}{625 x^{4}+50 x^{2} \ln \left (x \right )+\ln \left (x \right )^{2}+x -5}\)	\(29\)
risch	\(x -\frac {x^{2}}{625 x^{4}+50 x^{2} \ln \left (x \right )+\ln \left (x \right )^{2}+x -5}\)	\(29\)
parallelrisch	\(\frac {625 x^{5}+50 x^{3} \ln \left (x \right )+x \ln \left (x \right )^{2}-5 x}{625 x^{4}+50 x^{2} \ln \left (x \right )+\ln \left (x \right )^{2}+x -5}\)	\(45\)

Input:

int((ln(x)^4+100*x^2*ln(x)^3+(3750*x^4-10)*ln(x)^2+(62500*x^6+100*x^3-500* 
x^2+2*x)*ln(x)+390625*x^8+2500*x^5-6250*x^4+50*x^3+25)/(ln(x)^4+100*x^2*ln 
(x)^3+(3750*x^4+2*x-10)*ln(x)^2+(62500*x^6+100*x^3-500*x^2)*ln(x)+390625*x 
^8+1250*x^5-6250*x^4+x^2-10*x+25),x,method=_RETURNVERBOSE)
 

Output:

x-x^2/(625*x^4+50*x^2*ln(x)+ln(x)^2+x-5)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+\left (2 x-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+\left (-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+2 x+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx=\frac {625 \, x^{5} + 50 \, x^{3} \log \left (x\right ) + x \log \left (x\right )^{2} - 5 \, x}{625 \, x^{4} + 50 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2} + x - 5} \] Input:

integrate((log(x)^4+100*x^2*log(x)^3+(3750*x^4-10)*log(x)^2+(62500*x^6+100 
*x^3-500*x^2+2*x)*log(x)+390625*x^8+2500*x^5-6250*x^4+50*x^3+25)/(log(x)^4 
+100*x^2*log(x)^3+(3750*x^4+2*x-10)*log(x)^2+(62500*x^6+100*x^3-500*x^2)*l 
og(x)+390625*x^8+1250*x^5-6250*x^4+x^2-10*x+25),x, algorithm="fricas")
 

Output:

(625*x^5 + 50*x^3*log(x) + x*log(x)^2 - 5*x)/(625*x^4 + 50*x^2*log(x) + lo 
g(x)^2 + x - 5)
 

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+\left (2 x-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+\left (-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+2 x+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx=- \frac {x^{2}}{625 x^{4} + 50 x^{2} \log {\left (x \right )} + x + \log {\left (x \right )}^{2} - 5} + x \] Input:

integrate((ln(x)**4+100*x**2*ln(x)**3+(3750*x**4-10)*ln(x)**2+(62500*x**6+ 
100*x**3-500*x**2+2*x)*ln(x)+390625*x**8+2500*x**5-6250*x**4+50*x**3+25)/( 
ln(x)**4+100*x**2*ln(x)**3+(3750*x**4+2*x-10)*ln(x)**2+(62500*x**6+100*x** 
3-500*x**2)*ln(x)+390625*x**8+1250*x**5-6250*x**4+x**2-10*x+25),x)
 

Output:

-x**2/(625*x**4 + 50*x**2*log(x) + x + log(x)**2 - 5) + x
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+\left (2 x-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+\left (-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+2 x+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx=\frac {625 \, x^{5} + 50 \, x^{3} \log \left (x\right ) + x \log \left (x\right )^{2} - 5 \, x}{625 \, x^{4} + 50 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2} + x - 5} \] Input:

integrate((log(x)^4+100*x^2*log(x)^3+(3750*x^4-10)*log(x)^2+(62500*x^6+100 
*x^3-500*x^2+2*x)*log(x)+390625*x^8+2500*x^5-6250*x^4+50*x^3+25)/(log(x)^4 
+100*x^2*log(x)^3+(3750*x^4+2*x-10)*log(x)^2+(62500*x^6+100*x^3-500*x^2)*l 
og(x)+390625*x^8+1250*x^5-6250*x^4+x^2-10*x+25),x, algorithm="maxima")
 

Output:

(625*x^5 + 50*x^3*log(x) + x*log(x)^2 - 5*x)/(625*x^4 + 50*x^2*log(x) + lo 
g(x)^2 + x - 5)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+\left (2 x-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+\left (-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+2 x+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx=x - \frac {x^{2}}{625 \, x^{4} + 50 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2} + x - 5} \] Input:

integrate((log(x)^4+100*x^2*log(x)^3+(3750*x^4-10)*log(x)^2+(62500*x^6+100 
*x^3-500*x^2+2*x)*log(x)+390625*x^8+2500*x^5-6250*x^4+50*x^3+25)/(log(x)^4 
+100*x^2*log(x)^3+(3750*x^4+2*x-10)*log(x)^2+(62500*x^6+100*x^3-500*x^2)*l 
og(x)+390625*x^8+1250*x^5-6250*x^4+x^2-10*x+25),x, algorithm="giac")
 

Output:

x - x^2/(625*x^4 + 50*x^2*log(x) + log(x)^2 + x - 5)
 

Mupad [B] (verification not implemented)

Time = 7.60 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+\left (2 x-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+\left (-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+2 x+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx=x-\frac {x^2}{625\,x^4+50\,x^2\,\ln \left (x\right )+x+{\ln \left (x\right )}^2-5} \] Input:

int((log(x)*(2*x - 500*x^2 + 100*x^3 + 62500*x^6) + log(x)^2*(3750*x^4 - 1 
0) + log(x)^4 + 100*x^2*log(x)^3 + 50*x^3 - 6250*x^4 + 2500*x^5 + 390625*x 
^8 + 25)/(log(x)^2*(2*x + 3750*x^4 - 10) - 10*x + log(x)^4 + log(x)*(100*x 
^3 - 500*x^2 + 62500*x^6) + 100*x^2*log(x)^3 + x^2 - 6250*x^4 + 1250*x^5 + 
 390625*x^8 + 25),x)
 

Output:

x - x^2/(x + 50*x^2*log(x) + log(x)^2 + 625*x^4 - 5)
 

Reduce [F]

\[ \int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+\left (2 x-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+\left (-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+2 x+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx =\text {Too large to display} \] Input:

int((log(x)^4+100*x^2*log(x)^3+(3750*x^4-10)*log(x)^2+(62500*x^6+100*x^3-5 
00*x^2+2*x)*log(x)+390625*x^8+2500*x^5-6250*x^4+50*x^3+25)/(log(x)^4+100*x 
^2*log(x)^3+(3750*x^4+2*x-10)*log(x)^2+(62500*x^6+100*x^3-500*x^2)*log(x)+ 
390625*x^8+1250*x^5-6250*x^4+x^2-10*x+25),x)
 

Output:

int(log(x)**4/(log(x)**4 + 100*log(x)**3*x**2 + 3750*log(x)**2*x**4 + 2*lo 
g(x)**2*x - 10*log(x)**2 + 62500*log(x)*x**6 + 100*log(x)*x**3 - 500*log(x 
)*x**2 + 390625*x**8 + 1250*x**5 - 6250*x**4 + x**2 - 10*x + 25),x) - 10*i 
nt(log(x)**2/(log(x)**4 + 100*log(x)**3*x**2 + 3750*log(x)**2*x**4 + 2*log 
(x)**2*x - 10*log(x)**2 + 62500*log(x)*x**6 + 100*log(x)*x**3 - 500*log(x) 
*x**2 + 390625*x**8 + 1250*x**5 - 6250*x**4 + x**2 - 10*x + 25),x) + 39062 
5*int(x**8/(log(x)**4 + 100*log(x)**3*x**2 + 3750*log(x)**2*x**4 + 2*log(x 
)**2*x - 10*log(x)**2 + 62500*log(x)*x**6 + 100*log(x)*x**3 - 500*log(x)*x 
**2 + 390625*x**8 + 1250*x**5 - 6250*x**4 + x**2 - 10*x + 25),x) + 2500*in 
t(x**5/(log(x)**4 + 100*log(x)**3*x**2 + 3750*log(x)**2*x**4 + 2*log(x)**2 
*x - 10*log(x)**2 + 62500*log(x)*x**6 + 100*log(x)*x**3 - 500*log(x)*x**2 
+ 390625*x**8 + 1250*x**5 - 6250*x**4 + x**2 - 10*x + 25),x) - 6250*int(x* 
*4/(log(x)**4 + 100*log(x)**3*x**2 + 3750*log(x)**2*x**4 + 2*log(x)**2*x - 
 10*log(x)**2 + 62500*log(x)*x**6 + 100*log(x)*x**3 - 500*log(x)*x**2 + 39 
0625*x**8 + 1250*x**5 - 6250*x**4 + x**2 - 10*x + 25),x) + 50*int(x**3/(lo 
g(x)**4 + 100*log(x)**3*x**2 + 3750*log(x)**2*x**4 + 2*log(x)**2*x - 10*lo 
g(x)**2 + 62500*log(x)*x**6 + 100*log(x)*x**3 - 500*log(x)*x**2 + 390625*x 
**8 + 1250*x**5 - 6250*x**4 + x**2 - 10*x + 25),x) + 100*int((log(x)**3*x* 
*2)/(log(x)**4 + 100*log(x)**3*x**2 + 3750*log(x)**2*x**4 + 2*log(x)**2*x 
- 10*log(x)**2 + 62500*log(x)*x**6 + 100*log(x)*x**3 - 500*log(x)*x**2 ...