Integrand size = 59, antiderivative size = 30 \[ \int \frac {2+e^{-50-25 e^{3 x}+5 x+e^{-50-25 e^{3 x}+5 x} x} \left (5 x+25 x^2-375 e^{3 x} x^2\right )}{5 x} \, dx=e^{e^{5 \left (-5 \left (2+e^{3 x}\right )+x\right )} x}-\frac {1}{5} \log \left (\frac {100}{x^2}\right ) \] Output:
exp(x*exp(-25*exp(3*x)+5*x-50))-1/5*ln(100/x^2)
Time = 0.55 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {2+e^{-50-25 e^{3 x}+5 x+e^{-50-25 e^{3 x}+5 x} x} \left (5 x+25 x^2-375 e^{3 x} x^2\right )}{5 x} \, dx=e^{e^{-50-25 e^{3 x}+5 x} x}+\frac {2 \log (x)}{5} \] Input:
Integrate[(2 + E^(-50 - 25*E^(3*x) + 5*x + E^(-50 - 25*E^(3*x) + 5*x)*x)*( 5*x + 25*x^2 - 375*E^(3*x)*x^2))/(5*x),x]
Output:
E^(E^(-50 - 25*E^(3*x) + 5*x)*x) + (2*Log[x])/5
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.
\(\displaystyle \int \frac {e^{e^{5 x-25 e^{3 x}-50} x+5 x-25 e^{3 x}-50} \left (-375 e^{3 x} x^2+25 x^2+5 x\right )+2}{5 x} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \frac {5 e^{e^{5 x-25 e^{3 x}-50} x+5 x-25 e^{3 x}-50} \left (-75 e^{3 x} x^2+5 x^2+x\right )+2}{x}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {1}{5} \int \left (\frac {2}{x}-5 e^{e^{5 x-25 e^{3 x}-50} x+5 x-25 e^{3 x}-50} \left (75 e^{3 x} x-5 x-1\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (5 \int e^{e^{5 x-25 e^{3 x}-50} x+5 x-25 e^{3 x}-50}dx+25 \int e^{e^{5 x-25 e^{3 x}-50} x+5 x-25 e^{3 x}-50} xdx-375 \int e^{e^{5 x-25 e^{3 x}-50} x+8 x-25 e^{3 x}-50} xdx+2 \log (x)\right )\) |
Input:
Int[(2 + E^(-50 - 25*E^(3*x) + 5*x + E^(-50 - 25*E^(3*x) + 5*x)*x)*(5*x + 25*x^2 - 375*E^(3*x)*x^2))/(5*x),x]
Output:
$Aborted
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70
method | result | size |
norman | \({\mathrm e}^{x \,{\mathrm e}^{-25 \,{\mathrm e}^{3 x}+5 x -50}}+\frac {2 \ln \left (x \right )}{5}\) | \(21\) |
risch | \({\mathrm e}^{x \,{\mathrm e}^{-25 \,{\mathrm e}^{3 x}+5 x -50}}+\frac {2 \ln \left (x \right )}{5}\) | \(21\) |
parallelrisch | \({\mathrm e}^{x \,{\mathrm e}^{-25 \,{\mathrm e}^{3 x}+5 x -50}}+\frac {2 \ln \left (x \right )}{5}\) | \(21\) |
parts | \({\mathrm e}^{x \,{\mathrm e}^{-25 \,{\mathrm e}^{3 x}+5 x -50}}+\frac {2 \ln \left (x \right )}{5}\) | \(21\) |
Input:
int(1/5*((-375*x^2*exp(3*x)+25*x^2+5*x)*exp(-25*exp(3*x)+5*x-50)*exp(x*exp (-25*exp(3*x)+5*x-50))+2)/x,x,method=_RETURNVERBOSE)
Output:
exp(x*exp(-25*exp(3*x)+5*x-50))+2/5*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (24) = 48\).
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int \frac {2+e^{-50-25 e^{3 x}+5 x+e^{-50-25 e^{3 x}+5 x} x} \left (5 x+25 x^2-375 e^{3 x} x^2\right )}{5 x} \, dx=\frac {1}{5} \, {\left (2 \, e^{\left (5 \, x - 25 \, e^{\left (3 \, x\right )} - 50\right )} \log \left (x\right ) + 5 \, e^{\left (x e^{\left (5 \, x - 25 \, e^{\left (3 \, x\right )} - 50\right )} + 5 \, x - 25 \, e^{\left (3 \, x\right )} - 50\right )}\right )} e^{\left (-5 \, x + 25 \, e^{\left (3 \, x\right )} + 50\right )} \] Input:
integrate(1/5*((-375*x^2*exp(3*x)+25*x^2+5*x)*exp(-25*exp(3*x)+5*x-50)*exp (x*exp(-25*exp(3*x)+5*x-50))+2)/x,x, algorithm="fricas")
Output:
1/5*(2*e^(5*x - 25*e^(3*x) - 50)*log(x) + 5*e^(x*e^(5*x - 25*e^(3*x) - 50) + 5*x - 25*e^(3*x) - 50))*e^(-5*x + 25*e^(3*x) + 50)
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {2+e^{-50-25 e^{3 x}+5 x+e^{-50-25 e^{3 x}+5 x} x} \left (5 x+25 x^2-375 e^{3 x} x^2\right )}{5 x} \, dx=e^{x e^{5 x - 25 e^{3 x} - 50}} + \frac {2 \log {\left (x \right )}}{5} \] Input:
integrate(1/5*((-375*x**2*exp(3*x)+25*x**2+5*x)*exp(-25*exp(3*x)+5*x-50)*e xp(x*exp(-25*exp(3*x)+5*x-50))+2)/x,x)
Output:
exp(x*exp(5*x - 25*exp(3*x) - 50)) + 2*log(x)/5
Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {2+e^{-50-25 e^{3 x}+5 x+e^{-50-25 e^{3 x}+5 x} x} \left (5 x+25 x^2-375 e^{3 x} x^2\right )}{5 x} \, dx=e^{\left (x e^{\left (5 \, x - 25 \, e^{\left (3 \, x\right )} - 50\right )}\right )} + \frac {2}{5} \, \log \left (x\right ) \] Input:
integrate(1/5*((-375*x^2*exp(3*x)+25*x^2+5*x)*exp(-25*exp(3*x)+5*x-50)*exp (x*exp(-25*exp(3*x)+5*x-50))+2)/x,x, algorithm="maxima")
Output:
e^(x*e^(5*x - 25*e^(3*x) - 50)) + 2/5*log(x)
\[ \int \frac {2+e^{-50-25 e^{3 x}+5 x+e^{-50-25 e^{3 x}+5 x} x} \left (5 x+25 x^2-375 e^{3 x} x^2\right )}{5 x} \, dx=\int { -\frac {5 \, {\left (75 \, x^{2} e^{\left (3 \, x\right )} - 5 \, x^{2} - x\right )} e^{\left (x e^{\left (5 \, x - 25 \, e^{\left (3 \, x\right )} - 50\right )} + 5 \, x - 25 \, e^{\left (3 \, x\right )} - 50\right )} - 2}{5 \, x} \,d x } \] Input:
integrate(1/5*((-375*x^2*exp(3*x)+25*x^2+5*x)*exp(-25*exp(3*x)+5*x-50)*exp (x*exp(-25*exp(3*x)+5*x-50))+2)/x,x, algorithm="giac")
Output:
integrate(-1/5*(5*(75*x^2*e^(3*x) - 5*x^2 - x)*e^(x*e^(5*x - 25*e^(3*x) - 50) + 5*x - 25*e^(3*x) - 50) - 2)/x, x)
Time = 3.48 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \frac {2+e^{-50-25 e^{3 x}+5 x+e^{-50-25 e^{3 x}+5 x} x} \left (5 x+25 x^2-375 e^{3 x} x^2\right )}{5 x} \, dx={\mathrm {e}}^{x\,{\mathrm {e}}^{-25\,{\mathrm {e}}^{3\,x}}\,{\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{-50}}+\frac {2\,\ln \left (x\right )}{5} \] Input:
int(((exp(5*x - 25*exp(3*x) - 50)*exp(x*exp(5*x - 25*exp(3*x) - 50))*(5*x - 375*x^2*exp(3*x) + 25*x^2))/5 + 2/5)/x,x)
Output:
exp(x*exp(-25*exp(3*x))*exp(5*x)*exp(-50)) + (2*log(x))/5
\[ \int \frac {2+e^{-50-25 e^{3 x}+5 x+e^{-50-25 e^{3 x}+5 x} x} \left (5 x+25 x^2-375 e^{3 x} x^2\right )}{5 x} \, dx=\frac {5 \left (\int \frac {e^{\frac {5 e^{25 e^{3 x}} e^{50} x +e^{5 x} x}{e^{25 e^{3 x}} e^{50}}}}{e^{25 e^{3 x}}}d x \right )-375 \left (\int \frac {e^{\frac {8 e^{25 e^{3 x}} e^{50} x +e^{5 x} x}{e^{25 e^{3 x}} e^{50}}} x}{e^{25 e^{3 x}}}d x \right )+25 \left (\int \frac {e^{\frac {5 e^{25 e^{3 x}} e^{50} x +e^{5 x} x}{e^{25 e^{3 x}} e^{50}}} x}{e^{25 e^{3 x}}}d x \right )+2 \,\mathrm {log}\left (x \right ) e^{50}}{5 e^{50}} \] Input:
int(1/5*((-375*x^2*exp(3*x)+25*x^2+5*x)*exp(-25*exp(3*x)+5*x-50)*exp(x*exp (-25*exp(3*x)+5*x-50))+2)/x,x)
Output:
(5*int(e**((5*e**(25*e**(3*x))*e**50*x + e**(5*x)*x)/(e**(25*e**(3*x))*e** 50))/e**(25*e**(3*x)),x) - 375*int((e**((8*e**(25*e**(3*x))*e**50*x + e**( 5*x)*x)/(e**(25*e**(3*x))*e**50))*x)/e**(25*e**(3*x)),x) + 25*int((e**((5* e**(25*e**(3*x))*e**50*x + e**(5*x)*x)/(e**(25*e**(3*x))*e**50))*x)/e**(25 *e**(3*x)),x) + 2*log(x)*e**50)/(5*e**50)