\(\int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} (160 x^2-160 x^3+40 x^4+(180 x^2-180 x^3+45 x^4) \log (x)+x^{-\frac {x}{-10+5 x}} (-20+22 x-6 x^2+2 x \log (x)))}{20 x^2-20 x^3+5 x^4} \, dx\) [413]

Optimal result

Integrand size = 116, antiderivative size = 28 \[ \int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{20 x^2-20 x^3+5 x^4} \, dx=e^{3-x+x^{-1+\frac {x}{5 (2-x)}}+9 x \log (x)} \] Output:

exp(9*x*ln(x)-x+exp(1/5*x*ln(x)/(2-x))/x+3)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{20 x^2-20 x^3+5 x^4} \, dx=e^{3-x+x^{-1-\frac {x}{5 (-2+x)}}} x^{9 x} \] Input:

Integrate[(E^((3*x - x^2 + x^(-(x/(-10 + 5*x))) + 9*x^2*Log[x])/x)*(160*x^ 
2 - 160*x^3 + 40*x^4 + (180*x^2 - 180*x^3 + 45*x^4)*Log[x] + (-20 + 22*x - 
 6*x^2 + 2*x*Log[x])/x^(x/(-10 + 5*x))))/(20*x^2 - 20*x^3 + 5*x^4),x]
 

Output:

E^(3 - x + x^(-1 - x/(5*(-2 + x))))*x^(9*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.

\(\displaystyle \int \frac {e^{\frac {x^{-\frac {x}{5 x-10}}-x^2+9 x^2 \log (x)+3 x}{x}} \left (40 x^4-160 x^3+160 x^2+x^{-\frac {x}{5 x-10}} \left (-6 x^2+22 x+2 x \log (x)-20\right )+\left (45 x^4-180 x^3+180 x^2\right ) \log (x)\right )}{5 x^4-20 x^3+20 x^2} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {e^{\frac {x^{-\frac {x}{5 x-10}}-x^2+9 x^2 \log (x)+3 x}{x}} \left (40 x^4-160 x^3+160 x^2+x^{-\frac {x}{5 x-10}} \left (-6 x^2+22 x+2 x \log (x)-20\right )+\left (45 x^4-180 x^3+180 x^2\right ) \log (x)\right )}{x^2 \left (5 x^2-20 x+20\right )}dx\)

\(\Big \downarrow \) 7277

\(\displaystyle 20 \int \frac {e^{\frac {x^{\frac {x}{5 (2-x)}}-x^2+3 x}{x}} x^{9 x-2} \left (-2 \left (3 x^2-\log (x) x-11 x+10\right ) x^{\frac {x}{5 (2-x)}}+40 x^4-160 x^3+160 x^2+45 \left (x^4-4 x^3+4 x^2\right ) \log (x)\right )}{100 (2-x)^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int \frac {e^{\frac {x^{\frac {x}{5 (2-x)}}-x^2+3 x}{x}} x^{9 x-2} \left (-2 \left (3 x^2-\log (x) x-11 x+10\right ) x^{\frac {x}{5 (2-x)}}+40 x^4-160 x^3+160 x^2+45 \left (x^4-4 x^3+4 x^2\right ) \log (x)\right )}{(2-x)^2}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {1}{5} \int \frac {e^{x^{\frac {x}{5 (2-x)}-1}-x+3} x^{9 x-2} \left (-2 \left (3 x^2-\log (x) x-11 x+10\right ) x^{\frac {x}{5 (2-x)}}+40 x^4-160 x^3+160 x^2+45 \left (x^4-4 x^3+4 x^2\right ) \log (x)\right )}{(2-x)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{5} \int \left (5 e^{x^{\frac {x}{5 (2-x)}-1}-x+3} x^{9 x} (9 \log (x)+8)-\frac {2 e^{x^{\frac {x}{5 (2-x)}-1}-x+3} x^{\frac {x}{10-5 x}+9 x-2} \left (3 x^2-\log (x) x-11 x+10\right )}{(x-2)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} \left (40 \int e^{x^{\frac {x}{5 (2-x)}-1}-x+3} x^{9 x}dx-6 \int e^{x^{\frac {x}{5 (2-x)}-1}-x+3} x^{\frac {x}{10-5 x}+9 x-2}dx-2 \int \frac {e^{x^{\frac {x}{5 (2-x)}-1}-x+3} x^{\frac {x}{10-5 x}+9 x-2}}{x-2}dx-45 \int \frac {\int e^{x^{\frac {x}{10-5 x}-1}-x+3} x^{9 x}dx}{x}dx-2 \int \frac {\int \frac {e^{x^{\frac {x}{10-5 x}-1}-x+3} x^{\left (9+\frac {1}{10-5 x}\right ) x-1}}{(x-2)^2}dx}{x}dx+45 \log (x) \int e^{x^{\frac {x}{5 (2-x)}-1}-x+3} x^{9 x}dx+2 \log (x) \int \frac {e^{x^{\frac {x}{5 (2-x)}-1}-x+3} x^{\frac {x}{10-5 x}+9 x-1}}{(x-2)^2}dx\right )\)

Input:

Int[(E^((3*x - x^2 + x^(-(x/(-10 + 5*x))) + 9*x^2*Log[x])/x)*(160*x^2 - 16 
0*x^3 + 40*x^4 + (180*x^2 - 180*x^3 + 45*x^4)*Log[x] + (-20 + 22*x - 6*x^2 
 + 2*x*Log[x])/x^(x/(-10 + 5*x))))/(20*x^2 - 20*x^3 + 5*x^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 7.52 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14

method	result	size
risch	\(x^{9 x} {\mathrm e}^{-\frac {x^{2}-x^{-\frac {x}{5 \left (-2+x \right )}}-3 x}{x}}\)	\(32\)
parallelrisch	\({\mathrm e}^{\frac {{\mathrm e}^{-\frac {\ln \left (x \right ) x}{5 \left (-2+x \right )}}+9 x^{2} \ln \left (x \right )-x^{2}+3 x}{x}}\)	\(33\)

Input:

int(((2*x*ln(x)-6*x^2+22*x-20)*exp(-x*ln(x)/(5*x-10))+(45*x^4-180*x^3+180* 
x^2)*ln(x)+40*x^4-160*x^3+160*x^2)*exp((exp(-x*ln(x)/(5*x-10))+9*x^2*ln(x) 
-x^2+3*x)/x)/(5*x^4-20*x^3+20*x^2),x,method=_RETURNVERBOSE)
 

Output:

x^(9*x)*exp(-(x^2-x^(-1/5*x/(-2+x))-3*x)/x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{20 x^2-20 x^3+5 x^4} \, dx=e^{\left (\frac {{\left (9 \, x^{2} \log \left (x\right ) - x^{2} + 3 \, x\right )} x^{\frac {x}{5 \, {\left (x - 2\right )}}} + 1}{x x^{\frac {x}{5 \, {\left (x - 2\right )}}}}\right )} \] Input:

integrate(((2*x*log(x)-6*x^2+22*x-20)*exp(-x*log(x)/(5*x-10))+(45*x^4-180* 
x^3+180*x^2)*log(x)+40*x^4-160*x^3+160*x^2)*exp((exp(-x*log(x)/(5*x-10))+9 
*x^2*log(x)-x^2+3*x)/x)/(5*x^4-20*x^3+20*x^2),x, algorithm="fricas")
 

Output:

e^(((9*x^2*log(x) - x^2 + 3*x)*x^(1/5*x/(x - 2)) + 1)/(x*x^(1/5*x/(x - 2)) 
))
 

Sympy [A] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{20 x^2-20 x^3+5 x^4} \, dx=e^{\frac {9 x^{2} \log {\left (x \right )} - x^{2} + 3 x + e^{- \frac {x \log {\left (x \right )}}{5 x - 10}}}{x}} \] Input:

integrate(((2*x*ln(x)-6*x**2+22*x-20)*exp(-x*ln(x)/(5*x-10))+(45*x**4-180* 
x**3+180*x**2)*ln(x)+40*x**4-160*x**3+160*x**2)*exp((exp(-x*ln(x)/(5*x-10) 
)+9*x**2*ln(x)-x**2+3*x)/x)/(5*x**4-20*x**3+20*x**2),x)
 

Output:

exp((9*x**2*log(x) - x**2 + 3*x + exp(-x*log(x)/(5*x - 10)))/x)
 

Maxima [F]

\[ \int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{20 x^2-20 x^3+5 x^4} \, dx=\int { \frac {{\left (40 \, x^{4} - 160 \, x^{3} + 160 \, x^{2} + 45 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} \log \left (x\right ) - \frac {2 \, {\left (3 \, x^{2} - x \log \left (x\right ) - 11 \, x + 10\right )}}{x^{\frac {x}{5 \, {\left (x - 2\right )}}}}\right )} e^{\left (\frac {9 \, x^{2} \log \left (x\right ) - x^{2} + 3 \, x + \frac {1}{x^{\frac {x}{5 \, {\left (x - 2\right )}}}}}{x}\right )}}{5 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )}} \,d x } \] Input:

integrate(((2*x*log(x)-6*x^2+22*x-20)*exp(-x*log(x)/(5*x-10))+(45*x^4-180* 
x^3+180*x^2)*log(x)+40*x^4-160*x^3+160*x^2)*exp((exp(-x*log(x)/(5*x-10))+9 
*x^2*log(x)-x^2+3*x)/x)/(5*x^4-20*x^3+20*x^2),x, algorithm="maxima")
 

Output:

1/5*integrate((40*x^4 - 160*x^3 + 160*x^2 + 45*(x^4 - 4*x^3 + 4*x^2)*log(x 
) - 2*(3*x^2 - x*log(x) - 11*x + 10)/x^(1/5*x/(x - 2)))*e^((9*x^2*log(x) - 
 x^2 + 3*x + 1/x^(1/5*x/(x - 2)))/x)/(x^4 - 4*x^3 + 4*x^2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{20 x^2-20 x^3+5 x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((2*x*log(x)-6*x^2+22*x-20)*exp(-x*log(x)/(5*x-10))+(45*x^4-180* 
x^3+180*x^2)*log(x)+40*x^4-160*x^3+160*x^2)*exp((exp(-x*log(x)/(5*x-10))+9 
*x^2*log(x)-x^2+3*x)/x)/(5*x^4-20*x^3+20*x^2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{-2880000,[1,8]%%%}+%%%{25920000,[1,7]%%%} / %%%{1600000,[0 
,6]%%%} E
 

Mupad [B] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{20 x^2-20 x^3+5 x^4} \, dx=x^{9\,x}\,{\mathrm {e}}^{\frac {1}{x^{\frac {x}{5\,x-10}+1}}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3 \] Input:

int((exp((3*x + exp(-(x*log(x))/(5*x - 10)) + 9*x^2*log(x) - x^2)/x)*(exp( 
-(x*log(x))/(5*x - 10))*(22*x + 2*x*log(x) - 6*x^2 - 20) + log(x)*(180*x^2 
 - 180*x^3 + 45*x^4) + 160*x^2 - 160*x^3 + 40*x^4))/(20*x^2 - 20*x^3 + 5*x 
^4),x)
 

Output:

x^(9*x)*exp(1/x^(x/(5*x - 10) + 1))*exp(-x)*exp(3)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{20 x^2-20 x^3+5 x^4} \, dx=\frac {x^{9 x} e^{\frac {1}{e^{\frac {\mathrm {log}\left (x \right ) x}{5 x -10}} x}} e^{3}}{e^{x}} \] Input:

int(((2*x*log(x)-6*x^2+22*x-20)*exp(-x*log(x)/(5*x-10))+(45*x^4-180*x^3+18 
0*x^2)*log(x)+40*x^4-160*x^3+160*x^2)*exp((exp(-x*log(x)/(5*x-10))+9*x^2*l 
og(x)-x^2+3*x)/x)/(5*x^4-20*x^3+20*x^2),x)
 

Output:

(x**(9*x)*e**(1/(e**((log(x)*x)/(5*x - 10))*x))*e**3)/e**x