Integrand size = 50, antiderivative size = 19 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=e^{-\frac {27}{2} e^{-x} (-25+x) x} x^2 \] Output:
x^2/exp(27/2/exp(x)*(x-25)*x)
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=e^{-\frac {27}{2} e^{-x} (-25+x) x} x^2 \] Input:
Integrate[(E^(-x - (-675*x + 27*x^2)/(2*E^x))*(4*E^x*x + 675*x^2 - 729*x^3 + 27*x^4))/2,x]
Output:
x^2/E^((27*(-25 + x)*x)/(2*E^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.
\(\displaystyle \int \frac {1}{2} e^{-\frac {1}{2} e^{-x} \left (27 x^2-675 x\right )-x} \left (27 x^4-729 x^3+675 x^2+4 e^x x\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int e^{\frac {27}{2} e^{-x} \left (25 x-x^2\right )-x} \left (27 x^4-729 x^3+675 x^2+4 e^x x\right )dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {1}{2} \int e^{-\frac {1}{2} e^{-x} x \left (27 x+2 e^x-675\right )} \left (27 x^4-729 x^3+675 x^2+4 e^x x\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (27 e^{-\frac {1}{2} e^{-x} x \left (27 x+2 e^x-675\right )} x^4-729 e^{-\frac {1}{2} e^{-x} x \left (27 x+2 e^x-675\right )} x^3+675 e^{-\frac {1}{2} e^{-x} x \left (27 x+2 e^x-675\right )} x^2+4 e^{x-\frac {1}{2} e^{-x} x \left (27 x+2 e^x-675\right )} x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (27 \int e^{-\frac {1}{2} e^{-x} x \left (27 x+2 e^x-675\right )} x^4dx-729 \int e^{-\frac {1}{2} e^{-x} x \left (27 x+2 e^x-675\right )} x^3dx+675 \int e^{-\frac {1}{2} e^{-x} x \left (27 x+2 e^x-675\right )} x^2dx+4 \int e^{-\frac {27}{2} e^{-x} (x-25) x} xdx\right )\) |
Input:
Int[(E^(-x - (-675*x + 27*x^2)/(2*E^x))*(4*E^x*x + 675*x^2 - 729*x^3 + 27* x^4))/2,x]
Output:
$Aborted
Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
risch | \(x^{2} {\mathrm e}^{-\frac {27 x \left (x -25\right ) {\mathrm e}^{-x}}{2}}\) | \(16\) |
parallelrisch | \(x^{2} {\mathrm e}^{-\frac {27 x \left (x -25\right ) {\mathrm e}^{-x}}{2}}\) | \(18\) |
norman | \(x^{2} {\mathrm e}^{-\frac {\left (27 x^{2}-675 x \right ) {\mathrm e}^{-x}}{2}}\) | \(23\) |
Input:
int(1/2*(4*exp(x)*x+27*x^4-729*x^3+675*x^2)/exp(x)/exp(1/2*(27*x^2-675*x)/ exp(x)),x,method=_RETURNVERBOSE)
Output:
x^2*exp(-27/2*x*(x-25)*exp(-x))
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=x^{2} e^{\left (-\frac {1}{2} \, {\left (27 \, x^{2} + 2 \, x e^{x} - 675 \, x\right )} e^{\left (-x\right )} + x\right )} \] Input:
integrate(1/2*(4*exp(x)*x+27*x^4-729*x^3+675*x^2)/exp(x)/exp(1/2*(27*x^2-6 75*x)/exp(x)),x, algorithm="fricas")
Output:
x^2*e^(-1/2*(27*x^2 + 2*x*e^x - 675*x)*e^(-x) + x)
Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=x^{2} e^{- \left (\frac {27 x^{2}}{2} - \frac {675 x}{2}\right ) e^{- x}} \] Input:
integrate(1/2*(4*exp(x)*x+27*x**4-729*x**3+675*x**2)/exp(x)/exp(1/2*(27*x* *2-675*x)/exp(x)),x)
Output:
x**2*exp(-(27*x**2/2 - 675*x/2)*exp(-x))
\[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=\int { \frac {1}{2} \, {\left (27 \, x^{4} - 729 \, x^{3} + 675 \, x^{2} + 4 \, x e^{x}\right )} e^{\left (-\frac {27}{2} \, {\left (x^{2} - 25 \, x\right )} e^{\left (-x\right )} - x\right )} \,d x } \] Input:
integrate(1/2*(4*exp(x)*x+27*x^4-729*x^3+675*x^2)/exp(x)/exp(1/2*(27*x^2-6 75*x)/exp(x)),x, algorithm="maxima")
Output:
1/2*integrate((27*x^4 - 729*x^3 + 675*x^2 + 4*x*e^x)*e^(-27/2*(x^2 - 25*x) *e^(-x) - x), x)
\[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=\int { \frac {1}{2} \, {\left (27 \, x^{4} - 729 \, x^{3} + 675 \, x^{2} + 4 \, x e^{x}\right )} e^{\left (-\frac {27}{2} \, {\left (x^{2} - 25 \, x\right )} e^{\left (-x\right )} - x\right )} \,d x } \] Input:
integrate(1/2*(4*exp(x)*x+27*x^4-729*x^3+675*x^2)/exp(x)/exp(1/2*(27*x^2-6 75*x)/exp(x)),x, algorithm="giac")
Output:
integrate(1/2*(27*x^4 - 729*x^3 + 675*x^2 + 4*x*e^x)*e^(-27/2*(x^2 - 25*x) *e^(-x) - x), x)
Time = 3.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=x^2\,{\mathrm {e}}^{\frac {675\,x\,{\mathrm {e}}^{-x}}{2}}\,{\mathrm {e}}^{-\frac {27\,x^2\,{\mathrm {e}}^{-x}}{2}} \] Input:
int(exp(-x)*exp(exp(-x)*((675*x)/2 - (27*x^2)/2))*(2*x*exp(x) + (675*x^2)/ 2 - (729*x^3)/2 + (27*x^4)/2),x)
Output:
x^2*exp((675*x*exp(-x))/2)*exp(-(27*x^2*exp(-x))/2)
Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=\frac {e^{\frac {675 x}{2 e^{x}}} x^{2}}{e^{\frac {27 x^{2}}{2 e^{x}}}} \] Input:
int(1/2*(4*exp(x)*x+27*x^4-729*x^3+675*x^2)/exp(x)/exp(1/2*(27*x^2-675*x)/ exp(x)),x)
Output:
(e**((675*x)/(2*e**x))*x**2)/e**((27*x**2)/(2*e**x))