Integrand size = 86, antiderivative size = 32 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\frac {e^2 ((-1+x) x+\log (x))}{5-\frac {3-x}{20-x}}} \] Output:
exp((ln(x)+(-1+x)*x)/(5-(3-x)/(-x+20))*exp(2))
Time = 3.70 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\frac {e^2 x \left (20-21 x+x^2\right )}{-97+4 x}} x^{\frac {e^2 (-20+x)}{-97+4 x}} \] Input:
Integrate[(E^((E^2*(20*x - 21*x^2 + x^3) + E^2*(-20 + x)*Log[x])/(-97 + 4* x))*(E^2*(1940 - 2117*x + 4078*x^2 - 375*x^3 + 8*x^4) - 17*E^2*x*Log[x]))/ (9409*x - 776*x^2 + 16*x^3),x]
Output:
E^((E^2*x*(20 - 21*x + x^2))/(-97 + 4*x))*x^((E^2*(-20 + x))/(-97 + 4*x))
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.
\(\displaystyle \int \frac {\left (e^2 \left (8 x^4-375 x^3+4078 x^2-2117 x+1940\right )-17 e^2 x \log (x)\right ) \exp \left (\frac {e^2 \left (x^3-21 x^2+20 x\right )+e^2 (x-20) \log (x)}{4 x-97}\right )}{16 x^3-776 x^2+9409 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (e^2 \left (8 x^4-375 x^3+4078 x^2-2117 x+1940\right )-17 e^2 x \log (x)\right ) \exp \left (\frac {e^2 \left (x^3-21 x^2+20 x\right )+e^2 (x-20) \log (x)}{4 x-97}\right )}{x \left (16 x^2-776 x+9409\right )}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 64 \int \frac {e^{-\frac {e^2 \left (x^3-21 x^2+20 x\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}-1} \left (e^2 \left (8 x^4-375 x^3+4078 x^2-2117 x+1940\right )-17 e^2 x \log (x)\right )}{64 (97-4 x)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {e^{-\frac {e^2 \left (x^3-21 x^2+20 x\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}-1} \left (e^2 \left (8 x^4-375 x^3+4078 x^2-2117 x+1940\right )-17 e^2 x \log (x)\right )}{(97-4 x)^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}-1} \left (8 x^4-375 x^3+4078 x^2-2117 x-17 x \log (x)+1940\right )}{(97-4 x)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1940 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}-1}}{(4 x-97)^2}+\frac {4078 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+1}}{(4 x-97)^2}-\frac {375 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+2}}{(4 x-97)^2}+\frac {8 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+3}}{(4 x-97)^2}-\frac {2117 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}}}{(4 x-97)^2}-\frac {17 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}} \log (x)}{(4 x-97)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 1940 \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}-1}}{(4 x-97)^2}dx+4078 \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+1}}{(4 x-97)^2}dx-375 \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+2}}{(4 x-97)^2}dx+8 \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+3}}{(4 x-97)^2}dx-2117 \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}}}{(4 x-97)^2}dx+17 \int \frac {\int \frac {e^{\frac {e^2 x \left (x^2-21 x+20\right )}{4 x-97}+2} x^{\frac {e^2 (x-20)}{4 x-97}}}{(97-4 x)^2}dx}{x}dx-17 \log (x) \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}}}{(4 x-97)^2}dx\) |
Input:
Int[(E^((E^2*(20*x - 21*x^2 + x^3) + E^2*(-20 + x)*Log[x])/(-97 + 4*x))*(E ^2*(1940 - 2117*x + 4078*x^2 - 375*x^3 + 8*x^4) - 17*E^2*x*Log[x]))/(9409* x - 776*x^2 + 16*x^3),x]
Output:
$Aborted
Time = 2.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75
method | result | size |
risch | \({\mathrm e}^{\frac {{\mathrm e}^{2} \left (x -20\right ) \left (x^{2}+\ln \left (x \right )-x \right )}{4 x -97}}\) | \(24\) |
parallelrisch | \({\mathrm e}^{\frac {{\mathrm e}^{2} \left (x^{3}+x \ln \left (x \right )-21 x^{2}-20 \ln \left (x \right )+20 x \right )}{4 x -97}}\) | \(32\) |
norman | \(\frac {4 x \,{\mathrm e}^{\frac {\left (x -20\right ) {\mathrm e}^{2} \ln \left (x \right )+\left (x^{3}-21 x^{2}+20 x \right ) {\mathrm e}^{2}}{4 x -97}}-97 \,{\mathrm e}^{\frac {\left (x -20\right ) {\mathrm e}^{2} \ln \left (x \right )+\left (x^{3}-21 x^{2}+20 x \right ) {\mathrm e}^{2}}{4 x -97}}}{4 x -97}\) | \(81\) |
Input:
int((-17*x*exp(2)*ln(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2))*exp(( (x-20)*exp(2)*ln(x)+(x^3-21*x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-776*x^2+94 09*x),x,method=_RETURNVERBOSE)
Output:
exp(exp(2)*(x-20)*(x^2+ln(x)-x)/(4*x-97))
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\left (\frac {{\left (x - 20\right )} e^{2} \log \left (x\right ) + {\left (x^{3} - 21 \, x^{2} + 20 \, x\right )} e^{2}}{4 \, x - 97}\right )} \] Input:
integrate((-17*x*exp(2)*log(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2) )*exp(((x-20)*exp(2)*log(x)+(x^3-21*x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-77 6*x^2+9409*x),x, algorithm="fricas")
Output:
e^(((x - 20)*e^2*log(x) + (x^3 - 21*x^2 + 20*x)*e^2)/(4*x - 97))
Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\frac {\left (x - 20\right ) e^{2} \log {\left (x \right )} + \left (x^{3} - 21 x^{2} + 20 x\right ) e^{2}}{4 x - 97}} \] Input:
integrate((-17*x*exp(2)*ln(x)+(8*x**4-375*x**3+4078*x**2-2117*x+1940)*exp( 2))*exp(((x-20)*exp(2)*ln(x)+(x**3-21*x**2+20*x)*exp(2))/(4*x-97))/(16*x** 3-776*x**2+9409*x),x)
Output:
exp(((x - 20)*exp(2)*log(x) + (x**3 - 21*x**2 + 20*x)*exp(2))/(4*x - 97))
Time = 18.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\left (\frac {1}{4} \, x^{2} e^{2} + \frac {13}{16} \, x e^{2} + \frac {1}{4} \, e^{2} \log \left (x\right ) + \frac {17 \, e^{2} \log \left (x\right )}{4 \, {\left (4 \, x - 97\right )}} + \frac {153357 \, e^{2}}{64 \, {\left (4 \, x - 97\right )}} + \frac {1581}{64} \, e^{2}\right )} \] Input:
integrate((-17*x*exp(2)*log(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2) )*exp(((x-20)*exp(2)*log(x)+(x^3-21*x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-77 6*x^2+9409*x),x, algorithm="maxima")
Output:
e^(1/4*x^2*e^2 + 13/16*x*e^2 + 1/4*e^2*log(x) + 17/4*e^2*log(x)/(4*x - 97) + 153357/64*e^2/(4*x - 97) + 1581/64*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (26) = 52\).
Time = 0.14 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.09 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\left (\frac {x^{3} e^{2}}{4 \, x - 97} - \frac {21 \, x^{2} e^{2}}{4 \, x - 97} + \frac {x e^{2} \log \left (x\right )}{4 \, x - 97} + \frac {20 \, x e^{2}}{4 \, x - 97} - \frac {20 \, e^{2} \log \left (x\right )}{4 \, x - 97}\right )} \] Input:
integrate((-17*x*exp(2)*log(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2) )*exp(((x-20)*exp(2)*log(x)+(x^3-21*x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-77 6*x^2+9409*x),x, algorithm="giac")
Output:
e^(x^3*e^2/(4*x - 97) - 21*x^2*e^2/(4*x - 97) + x*e^2*log(x)/(4*x - 97) + 20*x*e^2/(4*x - 97) - 20*e^2*log(x)/(4*x - 97))
Time = 3.55 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=\frac {{\mathrm {e}}^{\frac {20\,x\,{\mathrm {e}}^2}{4\,x-97}}\,{\mathrm {e}}^{\frac {x^3\,{\mathrm {e}}^2}{4\,x-97}}\,{\mathrm {e}}^{-\frac {21\,x^2\,{\mathrm {e}}^2}{4\,x-97}}}{x^{\frac {20\,{\mathrm {e}}^2-x\,{\mathrm {e}}^2}{4\,x-97}}} \] Input:
int((exp((exp(2)*(20*x - 21*x^2 + x^3) + exp(2)*log(x)*(x - 20))/(4*x - 97 ))*(exp(2)*(4078*x^2 - 2117*x - 375*x^3 + 8*x^4 + 1940) - 17*x*exp(2)*log( x)))/(9409*x - 776*x^2 + 16*x^3),x)
Output:
(exp((20*x*exp(2))/(4*x - 97))*exp((x^3*exp(2))/(4*x - 97))*exp(-(21*x^2*e xp(2))/(4*x - 97)))/x^((20*exp(2) - x*exp(2))/(4*x - 97))
Time = 0.58 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=\frac {e^{\frac {\mathrm {log}\left (x \right ) e^{2} x +e^{2} x^{3}}{4 x -97}}}{e^{\frac {20 \,\mathrm {log}\left (x \right ) e^{2}+21 e^{2} x^{2}-20 e^{2} x}{4 x -97}}} \] Input:
int((-17*x*exp(2)*log(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2))*exp( ((x-20)*exp(2)*log(x)+(x^3-21*x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-776*x^2+ 9409*x),x)
Output:
e**((log(x)*e**2*x + e**2*x**3)/(4*x - 97))/e**((20*log(x)*e**2 + 21*e**2* x**2 - 20*e**2*x)/(4*x - 97))