\(\int \frac {e^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x))}{16 x^5+8 x^6+x^7+(8 x^5+2 x^6) \log ^2(x)+x^5 \log ^4(x)} \, dx\) [1313]

Optimal result

Integrand size = 106, antiderivative size = 24 \[ \int \frac {e^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} \left (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x)\right )}{16 x^5+8 x^6+x^7+\left (8 x^5+2 x^6\right ) \log ^2(x)+x^5 \log ^4(x)} \, dx=\frac {2 e^{5+\frac {x}{\frac {4+x}{\log (x)}+\log (x)}}}{x^4} \] Output:

2/x^4*exp(x/((4+x)/ln(x)+ln(x))+5)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} \left (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x)\right )}{16 x^5+8 x^6+x^7+\left (8 x^5+2 x^6\right ) \log ^2(x)+x^5 \log ^4(x)} \, dx=2 e^5 x^{-4+\frac {x}{4+x+\log ^2(x)}} \] Input:

Integrate[(E^((20 + 5*x + x*Log[x] + 5*Log[x]^2)/(4 + x + Log[x]^2))*(-128 
 - 56*x - 6*x^2 + 8*x*Log[x] + (-64 - 18*x)*Log[x]^2 + 2*x*Log[x]^3 - 8*Lo 
g[x]^4))/(16*x^5 + 8*x^6 + x^7 + (8*x^5 + 2*x^6)*Log[x]^2 + x^5*Log[x]^4), 
x]
 

Output:

2*E^5*x^(-4 + x/(4 + 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.

\(\displaystyle \int \frac {e^{\frac {5 x+5 \log ^2(x)+x \log (x)+20}{x+\log ^2(x)+4}} \left (-6 x^2-56 x-8 \log ^4(x)+2 x \log ^3(x)+(-18 x-64) \log ^2(x)+8 x \log (x)-128\right )}{x^7+8 x^6+16 x^5+x^5 \log ^4(x)+\left (2 x^6+8 x^5\right ) \log ^2(x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{\frac {5 x+5 \log ^2(x)+x \log (x)+20}{x+\log ^2(x)+4}} \left (-6 x^2-56 x-8 \log ^4(x)+2 x \log ^3(x)+(-18 x-64) \log ^2(x)+8 x \log (x)-128\right )}{x^5 \left (x+\log ^2(x)+4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {8 e^{\frac {5 x+5 \log ^2(x)+x \log (x)+20}{x+\log ^2(x)+4}}}{x^5}+\frac {2 e^{\frac {5 x+5 \log ^2(x)+x \log (x)+20}{x+\log ^2(x)+4}} (\log (x)-1)}{x^4 \left (x+\log ^2(x)+4\right )}-\frac {2 e^{\frac {5 x+5 \log ^2(x)+x \log (x)+20}{x+\log ^2(x)+4}} (-2 x+x \log (x)-8)}{x^4 \left (x+\log ^2(x)+4\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -8 \int \frac {e^{\frac {5 \log ^2(x)+x \log (x)+5 x+20}{\log ^2(x)+x+4}}}{x^5}dx+16 \int \frac {e^{\frac {5 \log ^2(x)+x \log (x)+5 x+20}{\log ^2(x)+x+4}}}{x^4 \left (\log ^2(x)+x+4\right )^2}dx-2 \int \frac {e^{\frac {5 \log ^2(x)+x \log (x)+5 x+20}{\log ^2(x)+x+4}}}{x^4 \left (\log ^2(x)+x+4\right )}dx+2 \int \frac {e^{\frac {5 \log ^2(x)+x \log (x)+5 x+20}{\log ^2(x)+x+4}} \log (x)}{x^4 \left (\log ^2(x)+x+4\right )}dx+4 \int \frac {e^{\frac {5 \log ^2(x)+x \log (x)+5 x+20}{\log ^2(x)+x+4}}}{x^3 \left (\log ^2(x)+x+4\right )^2}dx-2 \int \frac {e^{\frac {5 \log ^2(x)+x \log (x)+5 x+20}{\log ^2(x)+x+4}} \log (x)}{x^3 \left (\log ^2(x)+x+4\right )^2}dx\)

Input:

Int[(E^((20 + 5*x + x*Log[x] + 5*Log[x]^2)/(4 + x + Log[x]^2))*(-128 - 56* 
x - 6*x^2 + 8*x*Log[x] + (-64 - 18*x)*Log[x]^2 + 2*x*Log[x]^3 - 8*Log[x]^4 
))/(16*x^5 + 8*x^6 + x^7 + (8*x^5 + 2*x^6)*Log[x]^2 + x^5*Log[x]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.34 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33

method	result	size
risch	\(\frac {2 \,{\mathrm e}^{\frac {5 \ln \left (x \right )^{2}+x \ln \left (x \right )+20+5 x}{\ln \left (x \right )^{2}+4+x}}}{x^{4}}\)	\(32\)
parallelrisch	\(\frac {2 \,{\mathrm e}^{\frac {5 \ln \left (x \right )^{2}+x \ln \left (x \right )+20+5 x}{\ln \left (x \right )^{2}+4+x}}}{x^{4}}\)	\(32\)

Input:

int((-8*ln(x)^4+2*x*ln(x)^3+(-18*x-64)*ln(x)^2+8*x*ln(x)-6*x^2-56*x-128)*e 
xp((5*ln(x)^2+x*ln(x)+20+5*x)/(ln(x)^2+4+x))/(x^5*ln(x)^4+(2*x^6+8*x^5)*ln 
(x)^2+x^7+8*x^6+16*x^5),x,method=_RETURNVERBOSE)
 

Output:

2/x^4*exp((5*ln(x)^2+x*ln(x)+20+5*x)/(ln(x)^2+4+x))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} \left (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x)\right )}{16 x^5+8 x^6+x^7+\left (8 x^5+2 x^6\right ) \log ^2(x)+x^5 \log ^4(x)} \, dx=\frac {2 \, e^{\left (\frac {x \log \left (x\right ) + 5 \, \log \left (x\right )^{2} + 5 \, x + 20}{\log \left (x\right )^{2} + x + 4}\right )}}{x^{4}} \] Input:

integrate((-8*log(x)^4+2*x*log(x)^3+(-18*x-64)*log(x)^2+8*x*log(x)-6*x^2-5 
6*x-128)*exp((5*log(x)^2+x*log(x)+20+5*x)/(log(x)^2+4+x))/(x^5*log(x)^4+(2 
*x^6+8*x^5)*log(x)^2+x^7+8*x^6+16*x^5),x, algorithm="fricas")
 

Output:

2*e^((x*log(x) + 5*log(x)^2 + 5*x + 20)/(log(x)^2 + x + 4))/x^4
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} \left (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x)\right )}{16 x^5+8 x^6+x^7+\left (8 x^5+2 x^6\right ) \log ^2(x)+x^5 \log ^4(x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-8*ln(x)**4+2*x*ln(x)**3+(-18*x-64)*ln(x)**2+8*x*ln(x)-6*x**2-5 
6*x-128)*exp((5*ln(x)**2+x*ln(x)+20+5*x)/(ln(x)**2+4+x))/(x**5*ln(x)**4+(2 
*x**6+8*x**5)*ln(x)**2+x**7+8*x**6+16*x**5),x)
 

Output:

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

Maxima [F]

\[ \int \frac {e^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} \left (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x)\right )}{16 x^5+8 x^6+x^7+\left (8 x^5+2 x^6\right ) \log ^2(x)+x^5 \log ^4(x)} \, dx=\int { \frac {2 \, {\left (x \log \left (x\right )^{3} - 4 \, \log \left (x\right )^{4} - {\left (9 \, x + 32\right )} \log \left (x\right )^{2} - 3 \, x^{2} + 4 \, x \log \left (x\right ) - 28 \, x - 64\right )} e^{\left (\frac {x \log \left (x\right ) + 5 \, \log \left (x\right )^{2} + 5 \, x + 20}{\log \left (x\right )^{2} + x + 4}\right )}}{x^{5} \log \left (x\right )^{4} + x^{7} + 8 \, x^{6} + 16 \, x^{5} + 2 \, {\left (x^{6} + 4 \, x^{5}\right )} \log \left (x\right )^{2}} \,d x } \] Input:

integrate((-8*log(x)^4+2*x*log(x)^3+(-18*x-64)*log(x)^2+8*x*log(x)-6*x^2-5 
6*x-128)*exp((5*log(x)^2+x*log(x)+20+5*x)/(log(x)^2+4+x))/(x^5*log(x)^4+(2 
*x^6+8*x^5)*log(x)^2+x^7+8*x^6+16*x^5),x, algorithm="maxima")
 

Output:

2*integrate((x*log(x)^3 - 4*log(x)^4 - (9*x + 32)*log(x)^2 - 3*x^2 + 4*x*l 
og(x) - 28*x - 64)*e^((x*log(x) + 5*log(x)^2 + 5*x + 20)/(log(x)^2 + x + 4 
))/(x^5*log(x)^4 + x^7 + 8*x^6 + 16*x^5 + 2*(x^6 + 4*x^5)*log(x)^2), x)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} \left (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x)\right )}{16 x^5+8 x^6+x^7+\left (8 x^5+2 x^6\right ) \log ^2(x)+x^5 \log ^4(x)} \, dx=\frac {2 \, x^{\frac {x}{\log \left (x\right )^{2} + x + 4}} e^{5}}{x^{4}} \] Input:

integrate((-8*log(x)^4+2*x*log(x)^3+(-18*x-64)*log(x)^2+8*x*log(x)-6*x^2-5 
6*x-128)*exp((5*log(x)^2+x*log(x)+20+5*x)/(log(x)^2+4+x))/(x^5*log(x)^4+(2 
*x^6+8*x^5)*log(x)^2+x^7+8*x^6+16*x^5),x, algorithm="giac")
 

Output:

2*x^(x/(log(x)^2 + x + 4))*e^5/x^4
 

Mupad [B] (verification not implemented)

Time = 3.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46 \[ \int \frac {e^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} \left (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x)\right )}{16 x^5+8 x^6+x^7+\left (8 x^5+2 x^6\right ) \log ^2(x)+x^5 \log ^4(x)} \, dx=\frac {2\,x^{\frac {x}{{\ln \left (x\right )}^2+x+4}}\,{\mathrm {e}}^{\frac {20}{{\ln \left (x\right )}^2+x+4}}\,{\mathrm {e}}^{\frac {5\,{\ln \left (x\right )}^2}{{\ln \left (x\right )}^2+x+4}}\,{\mathrm {e}}^{\frac {5\,x}{{\ln \left (x\right )}^2+x+4}}}{x^4} \] Input:

int(-(exp((5*x + 5*log(x)^2 + x*log(x) + 20)/(x + log(x)^2 + 4))*(56*x - 2 
*x*log(x)^3 + 8*log(x)^4 - 8*x*log(x) + 6*x^2 + log(x)^2*(18*x + 64) + 128 
))/(log(x)^2*(8*x^5 + 2*x^6) + x^5*log(x)^4 + 16*x^5 + 8*x^6 + x^7),x)
 

Output:

(2*x^(x/(x + log(x)^2 + 4))*exp(20/(x + log(x)^2 + 4))*exp((5*log(x)^2)/(x 
 + log(x)^2 + 4))*exp((5*x)/(x + log(x)^2 + 4)))/x^4
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} \left (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x)\right )}{16 x^5+8 x^6+x^7+\left (8 x^5+2 x^6\right ) \log ^2(x)+x^5 \log ^4(x)} \, dx=\frac {2 e^{\frac {\mathrm {log}\left (x \right ) x}{\mathrm {log}\left (x \right )^{2}+x +4}} e^{5}}{x^{4}} \] Input:

int((-8*log(x)^4+2*x*log(x)^3+(-18*x-64)*log(x)^2+8*x*log(x)-6*x^2-56*x-12 
8)*exp((5*log(x)^2+x*log(x)+20+5*x)/(log(x)^2+4+x))/(x^5*log(x)^4+(2*x^6+8 
*x^5)*log(x)^2+x^7+8*x^6+16*x^5),x)
 

Output:

(2*e**((log(x)*x)/(log(x)**2 + x + 4))*e**5)/x**4