\(\int \frac {e^{\frac {25 x^4+(x+400 x^3+25 x^4) \log (x)}{25 x+(400+25 x) \log (x)}} (x+75 x^4+(2400 x^3+150 x^4) \log (x)+(16+19200 x^2+2400 x^3+75 x^4) \log ^2(x))}{25 x^2+(800 x+50 x^2) \log (x)+(6400+800 x+25 x^2) \log ^2(x)} \, dx\) [1326]

Optimal result

Integrand size = 117, antiderivative size = 25 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=1+e^{x \left (x^2+\frac {1}{25 \left (16+x+\frac {x}{\log (x)}\right )}\right )} \] Output:

1+exp((x^2+1/25/(x+16+x/ln(x)))*x)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(25)=50\).

Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.76 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=e^{x^3+\frac {x}{400+25 x}+\frac {x^4}{x+(16+x) \log (x)}} x^{-\frac {x^2+400 x^4+25 x^5}{25 (16+x) \log (x) (x+(16+x) \log (x))}} \] Input:

Integrate[(E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*x + (400 + 25*x 
)*Log[x]))*(x + 75*x^4 + (2400*x^3 + 150*x^4)*Log[x] + (16 + 19200*x^2 + 2 
400*x^3 + 75*x^4)*Log[x]^2))/(25*x^2 + (800*x + 50*x^2)*Log[x] + (6400 + 8 
00*x + 25*x^2)*Log[x]^2),x]
 

Output:

E^(x^3 + x/(400 + 25*x) + x^4/(x + (16 + x)*Log[x]))/x^((x^2 + 400*x^4 + 2 
5*x^5)/(25*(16 + x)*Log[x]*(x + (16 + x)*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.

\(\displaystyle \int \frac {\left (75 x^4+\left (150 x^4+2400 x^3\right ) \log (x)+\left (75 x^4+2400 x^3+19200 x^2+16\right ) \log ^2(x)+x\right ) \exp \left (\frac {25 x^4+\left (25 x^4+400 x^3+x\right ) \log (x)}{25 x+(25 x+400) \log (x)}\right )}{25 x^2+\left (25 x^2+800 x+6400\right ) \log ^2(x)+\left (50 x^2+800 x\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (75 x^4+\left (150 x^4+2400 x^3\right ) \log (x)+\left (75 x^4+2400 x^3+19200 x^2+16\right ) \log ^2(x)+x\right ) \exp \left (\frac {25 x^4+\left (25 x^4+400 x^3+x\right ) \log (x)}{25 (x+x \log (x)+16 \log (x))}\right )}{25 (x+x \log (x)+16 \log (x))^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{25} \int \frac {e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} x^{\frac {25 x^4+400 x^3+x}{25 (\log (x) x+x+16 \log (x))}} \left (75 x^4+x+\left (75 x^4+2400 x^3+19200 x^2+16\right ) \log ^2(x)+150 \left (x^4+16 x^3\right ) \log (x)\right )}{(\log (x) x+x+16 \log (x))^2}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {1}{25} \int \frac {e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} x^{\frac {x \left (25 x^3+400 x^2+1\right )}{25 (\log (x) x+x+16 \log (x))}} \left (75 x^4+x+\left (75 x^4+2400 x^3+19200 x^2+16\right ) \log ^2(x)+150 \left (x^4+16 x^3\right ) \log (x)\right )}{(\log (x) x+x+16 \log (x))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{25} \int \left (\frac {e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} \left (75 x^4+2400 x^3+19200 x^2+16\right ) x^{\frac {x \left (25 x^3+400 x^2+1\right )}{25 (\log (x) x+x+16 \log (x))}}}{(x+16)^2}-\frac {32 e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} x^{\frac {x \left (25 x^3+400 x^2+1\right )}{25 (\log (x) x+x+16 \log (x))}+1}}{(x+16)^2 (\log (x) x+x+16 \log (x))}+\frac {e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} \left (x^2+48 x+256\right ) x^{\frac {x \left (25 x^3+400 x^2+1\right )}{25 (\log (x) x+x+16 \log (x))}+1}}{(x+16)^2 (\log (x) x+x+16 \log (x))^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{25} \left (75 \int e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} x^{\frac {x \left (25 x^3+400 x^2+1\right )}{25 (\log (x) x+x+16 \log (x))}+2}dx+\int \frac {e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} x^{\frac {x \left (25 x^3+400 x^2+1\right )}{25 (\log (x) x+x+16 \log (x))}+1}}{(\log (x) x+x+16 \log (x))^2}dx-256 \int \frac {e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} x^{\frac {x \left (25 x^3+400 x^2+1\right )}{25 (\log (x) x+x+16 \log (x))}+1}}{(x+16)^2 (\log (x) x+x+16 \log (x))^2}dx+16 \int \frac {e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} x^{\frac {x \left (25 x^3+400 x^2+1\right )}{25 (\log (x) x+x+16 \log (x))}+1}}{(x+16) (\log (x) x+x+16 \log (x))^2}dx-32 \int \frac {e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} x^{\frac {x \left (25 x^3+400 x^2+1\right )}{25 (\log (x) x+x+16 \log (x))}+1}}{(x+16)^2 (\log (x) x+x+16 \log (x))}dx+16 \int \frac {e^{\frac {x^4}{\log (x) x+x+16 \log (x)}} x^{\frac {x \left (25 x^3+400 x^2+1\right )}{25 (\log (x) x+x+16 \log (x))}}}{(x+16)^2}dx\right )\)

Input:

Int[(E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*x + (400 + 25*x)*Log[ 
x]))*(x + 75*x^4 + (2400*x^3 + 150*x^4)*Log[x] + (16 + 19200*x^2 + 2400*x^ 
3 + 75*x^4)*Log[x]^2))/(25*x^2 + (800*x + 50*x^2)*Log[x] + (6400 + 800*x + 
 25*x^2)*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.99 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48

method	result	size
parallelrisch	\({\mathrm e}^{\frac {\left (25 x^{4}+400 x^{3}+x \right ) \ln \left (x \right )+25 x^{4}}{25 x \ln \left (x \right )+400 \ln \left (x \right )+25 x}}\)	\(37\)
risch	\({\mathrm e}^{\frac {x \left (25 x^{3} \ln \left (x \right )+400 x^{2} \ln \left (x \right )+25 x^{3}+\ln \left (x \right )\right )}{25 x \ln \left (x \right )+400 \ln \left (x \right )+25 x}}\)	\(39\)

Input:

int(((75*x^4+2400*x^3+19200*x^2+16)*ln(x)^2+(150*x^4+2400*x^3)*ln(x)+75*x^ 
4+x)*exp(((25*x^4+400*x^3+x)*ln(x)+25*x^4)/((25*x+400)*ln(x)+25*x))/((25*x 
^2+800*x+6400)*ln(x)^2+(50*x^2+800*x)*ln(x)+25*x^2),x,method=_RETURNVERBOS 
E)
 

Output:

exp(1/25/(x*ln(x)+16*ln(x)+x)*((25*x^4+400*x^3+x)*ln(x)+25*x^4))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=e^{\left (\frac {25 \, x^{4} + {\left (25 \, x^{4} + 400 \, x^{3} + x\right )} \log \left (x\right )}{25 \, {\left ({\left (x + 16\right )} \log \left (x\right ) + x\right )}}\right )} \] Input:

integrate(((75*x^4+2400*x^3+19200*x^2+16)*log(x)^2+(150*x^4+2400*x^3)*log( 
x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)*log(x)+25*x^4)/((25*x+400)*log(x)+25* 
x))/((25*x^2+800*x+6400)*log(x)^2+(50*x^2+800*x)*log(x)+25*x^2),x, algorit 
hm="fricas")
 

Output:

e^(1/25*(25*x^4 + (25*x^4 + 400*x^3 + x)*log(x))/((x + 16)*log(x) + x))
 

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=e^{\frac {25 x^{4} + \left (25 x^{4} + 400 x^{3} + x\right ) \log {\left (x \right )}}{25 x + \left (25 x + 400\right ) \log {\left (x \right )}}} \] Input:

integrate(((75*x**4+2400*x**3+19200*x**2+16)*ln(x)**2+(150*x**4+2400*x**3) 
*ln(x)+75*x**4+x)*exp(((25*x**4+400*x**3+x)*ln(x)+25*x**4)/((25*x+400)*ln( 
x)+25*x))/((25*x**2+800*x+6400)*ln(x)**2+(50*x**2+800*x)*ln(x)+25*x**2),x)
 

Output:

exp((25*x**4 + (25*x**4 + 400*x**3 + x)*log(x))/(25*x + (25*x + 400)*log(x 
)))
 

Maxima [F(-1)]

Timed out. \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=\text {Timed out} \] Input:

integrate(((75*x^4+2400*x^3+19200*x^2+16)*log(x)^2+(150*x^4+2400*x^3)*log( 
x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)*log(x)+25*x^4)/((25*x+400)*log(x)+25* 
x))/((25*x^2+800*x+6400)*log(x)^2+(50*x^2+800*x)*log(x)+25*x^2),x, algorit 
hm="maxima")
 

Output:

Timed out
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (23) = 46\).

Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.88 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=e^{\left (\frac {x^{4} \log \left (x\right )}{x \log \left (x\right ) + x + 16 \, \log \left (x\right )} + \frac {x^{4}}{x \log \left (x\right ) + x + 16 \, \log \left (x\right )} + \frac {16 \, x^{3} \log \left (x\right )}{x \log \left (x\right ) + x + 16 \, \log \left (x\right )} + \frac {x \log \left (x\right )}{25 \, {\left (x \log \left (x\right ) + x + 16 \, \log \left (x\right )\right )}}\right )} \] Input:

integrate(((75*x^4+2400*x^3+19200*x^2+16)*log(x)^2+(150*x^4+2400*x^3)*log( 
x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)*log(x)+25*x^4)/((25*x+400)*log(x)+25* 
x))/((25*x^2+800*x+6400)*log(x)^2+(50*x^2+800*x)*log(x)+25*x^2),x, algorit 
hm="giac")
 

Output:

e^(x^4*log(x)/(x*log(x) + x + 16*log(x)) + x^4/(x*log(x) + x + 16*log(x)) 
+ 16*x^3*log(x)/(x*log(x) + x + 16*log(x)) + 1/25*x*log(x)/(x*log(x) + x + 
 16*log(x)))
 

Mupad [B] (verification not implemented)

Time = 3.84 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=x^{\frac {x^4+16\,x^3}{x+16\,\ln \left (x\right )+x\,\ln \left (x\right )}+\frac {x}{25\,x+400\,\ln \left (x\right )+25\,x\,\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {25\,x^4}{25\,x+400\,\ln \left (x\right )+25\,x\,\ln \left (x\right )}} \] Input:

int((exp((log(x)*(x + 400*x^3 + 25*x^4) + 25*x^4)/(25*x + log(x)*(25*x + 4 
00)))*(x + log(x)*(2400*x^3 + 150*x^4) + log(x)^2*(19200*x^2 + 2400*x^3 + 
75*x^4 + 16) + 75*x^4))/(log(x)^2*(800*x + 25*x^2 + 6400) + log(x)*(800*x 
+ 50*x^2) + 25*x^2),x)
 

Output:

x^((16*x^3 + x^4)/(x + 16*log(x) + x*log(x)) + x/(25*x + 400*log(x) + 25*x 
*log(x)))*exp((25*x^4)/(25*x + 400*log(x) + 25*x*log(x)))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=e^{\frac {25 \,\mathrm {log}\left (x \right ) x^{4}+400 \,\mathrm {log}\left (x \right ) x^{3}+\mathrm {log}\left (x \right ) x +25 x^{4}}{25 \,\mathrm {log}\left (x \right ) x +400 \,\mathrm {log}\left (x \right )+25 x}} \] Input:

int(((75*x^4+2400*x^3+19200*x^2+16)*log(x)^2+(150*x^4+2400*x^3)*log(x)+75* 
x^4+x)*exp(((25*x^4+400*x^3+x)*log(x)+25*x^4)/((25*x+400)*log(x)+25*x))/(( 
25*x^2+800*x+6400)*log(x)^2+(50*x^2+800*x)*log(x)+25*x^2),x)
 

Output:

e**((25*log(x)*x**4 + 400*log(x)*x**3 + log(x)*x + 25*x**4)/(25*log(x)*x + 
 400*log(x) + 25*x))