\(\int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} (-675 x+27 x^2)} (4 e^x x+675 x^2-729 x^3+27 x^4) \, dx\) [7994]

Optimal result

Integrand size = 50, antiderivative size = 19 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=e^{-\frac {27}{2} e^{-x} (-25+x) x} x^2 \]

[Out]

x^2/exp(27/2/exp(x)*(x-25)*x)

Rubi [F]

\[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=\int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx \]

[In]

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]

[Out]

2*Defer[Int][x/E^((27*(-25 + x)*x)/(2*E^x)), x] + (675*Defer[Int][x^2/E^((x*(-675 + 2*E^x + 27*x))/(2*E^x)), x
])/2 - (729*Defer[Int][x^3/E^((x*(-675 + 2*E^x + 27*x))/(2*E^x)), x])/2 + (27*Defer[Int][x^4/E^((x*(-675 + 2*E
^x + 27*x))/(2*E^x)), x])/2

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int 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 {1}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx \\ & = \frac {1}{2} \int \left (4 e^{x-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x+675 e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^2-729 e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^3+27 e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^4\right ) \, dx \\ & = 2 \int e^{x-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x \, dx+\frac {27}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^4 \, dx+\frac {675}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^2 \, dx-\frac {729}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^3 \, dx \\ & = 2 \int e^{-\frac {27}{2} e^{-x} (-25+x) x} x \, dx+\frac {27}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^4 \, dx+\frac {675}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^2 \, dx-\frac {729}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^3 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=e^{-\frac {27}{2} e^{-x} (-25+x) x} x^2 \]

[In]

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]

[Out]

x^2/E^((27*(-25 + x)*x)/(2*E^x))

Maple [A] (verified)

Time = 0.14 (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	\(x^{2} {\mathrm e}^{-\frac {\left (27 x^{2}-675 x \right ) {\mathrm e}^{-x}}{2}}\)	\(23\)

[In]

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)

[Out]

x^2*exp(-27/2*x*(x-25)*exp(-x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=x^{2} e^{\left (-\frac {1}{2} \, {\left (27 \, x^{2} + 2 \, x e^{x} - 675 \, x\right )} e^{\left (-x\right )} + x\right )} \]

[In]

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, algorithm="fricas")

[Out]

x^2*e^(-1/2*(27*x^2 + 2*x*e^x - 675*x)*e^(-x) + x)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=x^{2} e^{- \left (\frac {27 x^{2}}{2} - \frac {675 x}{2}\right ) e^{- x}} \]

[In]

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)

[Out]

x**2*exp(-(27*x**2/2 - 675*x/2)*exp(-x))

Maxima [F]

\[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=\int { \frac {1}{2} \, {\left (27 \, x^{4} - 729 \, x^{3} + 675 \, x^{2} + 4 \, x e^{x}\right )} e^{\left (-\frac {27}{2} \, {\left (x^{2} - 25 \, x\right )} e^{\left (-x\right )} - x\right )} \,d x } \]

[In]

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, algorithm="maxima")

[Out]

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)

Giac [F]

\[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=\int { \frac {1}{2} \, {\left (27 \, x^{4} - 729 \, x^{3} + 675 \, x^{2} + 4 \, x e^{x}\right )} e^{\left (-\frac {27}{2} \, {\left (x^{2} - 25 \, x\right )} e^{\left (-x\right )} - x\right )} \,d x } \]

[In]

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, algorithm="giac")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 12.92 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx=x^2\,{\mathrm {e}}^{\frac {675\,x\,{\mathrm {e}}^{-x}}{2}}\,{\mathrm {e}}^{-\frac {27\,x^2\,{\mathrm {e}}^{-x}}{2}} \]

[In]

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)

[Out]

x^2*exp((675*x*exp(-x))/2)*exp(-(27*x^2*exp(-x))/2)