3.50.58 \(\int \frac {1}{5} e^{-x-\frac {2}{5} e^{-x} (5 e^{2 x}-4 x+5 e^x x)} (32 x^2-40 e^{2 x} x^2-32 x^3+e^x (40 x-40 x^2)) \, dx\)

Optimal. Leaf size=26 \[ 4 e^{-2 e^x-2 x+\frac {8 e^{-x} x}{5}} x^2 \]

________________________________________________________________________________________

Rubi [F]  time = 1.33, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{5} \exp \left (-x-\frac {2}{5} e^{-x} \left (5 e^{2 x}-4 x+5 e^x x\right )\right ) \left (32 x^2-40 e^{2 x} x^2-32 x^3+e^x \left (40 x-40 x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-x - (2*(5*E^(2*x) - 4*x + 5*E^x*x))/(5*E^x))*(32*x^2 - 40*E^(2*x)*x^2 - 32*x^3 + E^x*(40*x - 40*x^2))
)/5,x]

[Out]

8*Defer[Int][E^(-2*E^x - 2*x + (8*x)/(5*E^x))*x, x] + (32*Defer[Int][E^(-2*E^x - 3*x + (8*x)/(5*E^x))*x^2, x])
/5 - 8*Defer[Int][E^(-2*E^x - 2*x + (8*x)/(5*E^x))*x^2, x] - 8*Defer[Int][E^(-2*E^x - x + (8*x)/(5*E^x))*x^2,
x] - (32*Defer[Int][E^(-2*E^x - 3*x + (8*x)/(5*E^x))*x^3, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \exp \left (-x-\frac {2}{5} e^{-x} \left (5 e^{2 x}-4 x+5 e^x x\right )\right ) \left (32 x^2-40 e^{2 x} x^2-32 x^3+e^x \left (40 x-40 x^2\right )\right ) \, dx\\ &=\frac {1}{5} \int 8 e^{-2 e^x-3 x+\frac {8 e^{-x} x}{5}} x \left (-5 e^x (-1+x)-5 e^{2 x} x-4 (-1+x) x\right ) \, dx\\ &=\frac {8}{5} \int e^{-2 e^x-3 x+\frac {8 e^{-x} x}{5}} x \left (-5 e^x (-1+x)-5 e^{2 x} x-4 (-1+x) x\right ) \, dx\\ &=\frac {8}{5} \int \left (-5 e^{-2 e^x-2 x+\frac {8 e^{-x} x}{5}} (-1+x) x-5 e^{-2 e^x-x+\frac {8 e^{-x} x}{5}} x^2-4 e^{-2 e^x-3 x+\frac {8 e^{-x} x}{5}} (-1+x) x^2\right ) \, dx\\ &=-\left (\frac {32}{5} \int e^{-2 e^x-3 x+\frac {8 e^{-x} x}{5}} (-1+x) x^2 \, dx\right )-8 \int e^{-2 e^x-2 x+\frac {8 e^{-x} x}{5}} (-1+x) x \, dx-8 \int e^{-2 e^x-x+\frac {8 e^{-x} x}{5}} x^2 \, dx\\ &=-\left (\frac {32}{5} \int \left (-e^{-2 e^x-3 x+\frac {8 e^{-x} x}{5}} x^2+e^{-2 e^x-3 x+\frac {8 e^{-x} x}{5}} x^3\right ) \, dx\right )-8 \int e^{-2 e^x-x+\frac {8 e^{-x} x}{5}} x^2 \, dx-8 \int \left (-e^{-2 e^x-2 x+\frac {8 e^{-x} x}{5}} x+e^{-2 e^x-2 x+\frac {8 e^{-x} x}{5}} x^2\right ) \, dx\\ &=\frac {32}{5} \int e^{-2 e^x-3 x+\frac {8 e^{-x} x}{5}} x^2 \, dx-\frac {32}{5} \int e^{-2 e^x-3 x+\frac {8 e^{-x} x}{5}} x^3 \, dx+8 \int e^{-2 e^x-2 x+\frac {8 e^{-x} x}{5}} x \, dx-8 \int e^{-2 e^x-2 x+\frac {8 e^{-x} x}{5}} x^2 \, dx-8 \int e^{-2 e^x-x+\frac {8 e^{-x} x}{5}} x^2 \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.35, size = 26, normalized size = 1.00 \begin {gather*} 4 e^{-2 e^x-2 x+\frac {8 e^{-x} x}{5}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x - (2*(5*E^(2*x) - 4*x + 5*E^x*x))/(5*E^x))*(32*x^2 - 40*E^(2*x)*x^2 - 32*x^3 + E^x*(40*x - 40
*x^2)))/5,x]

[Out]

4*E^(-2*E^x - 2*x + (8*x)/(5*E^x))*x^2

________________________________________________________________________________________

fricas [A]  time = 0.94, size = 29, normalized size = 1.12 \begin {gather*} 4 \, x^{2} e^{\left (-\frac {1}{5} \, {\left (15 \, x e^{x} - 8 \, x + 10 \, e^{\left (2 \, x\right )}\right )} e^{\left (-x\right )} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-40*exp(x)^2*x^2+(-40*x^2+40*x)*exp(x)-32*x^3+32*x^2)/exp(x)/exp(1/5*(5*exp(x)^2+5*exp(x)*x-4*x
)/exp(x))^2,x, algorithm="fricas")

[Out]

4*x^2*e^(-1/5*(15*x*e^x - 8*x + 10*e^(2*x))*e^(-x) + x)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 31, normalized size = 1.19 \begin {gather*} 4 \, x^{2} e^{\left (-\frac {1}{5} \, {\left (5 \, x e^{x} - 8 \, x + 10 \, e^{\left (2 \, x\right )}\right )} e^{\left (-x\right )} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-40*exp(x)^2*x^2+(-40*x^2+40*x)*exp(x)-32*x^3+32*x^2)/exp(x)/exp(1/5*(5*exp(x)^2+5*exp(x)*x-4*x
)/exp(x))^2,x, algorithm="giac")

[Out]

4*x^2*e^(-1/5*(5*x*e^x - 8*x + 10*e^(2*x))*e^(-x) - x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 28, normalized size = 1.08

method	result	size
risch	\(4 x^{2} {\mathrm e}^{-\frac {2 \left (5 \,{\mathrm e}^{2 x}+5 \,{\mathrm e}^{x} x -4 x \right ) {\mathrm e}^{-x}}{5}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(-40*exp(x)^2*x^2+(-40*x^2+40*x)*exp(x)-32*x^3+32*x^2)/exp(x)/exp(1/5*(5*exp(x)^2+5*exp(x)*x-4*x)/exp(
x))^2,x,method=_RETURNVERBOSE)

[Out]

4*x^2*exp(-2/5*(5*exp(2*x)+5*exp(x)*x-4*x)*exp(-x))

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 21, normalized size = 0.81 \begin {gather*} 4 \, x^{2} e^{\left (\frac {8}{5} \, x e^{\left (-x\right )} - 2 \, x - 2 \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-40*exp(x)^2*x^2+(-40*x^2+40*x)*exp(x)-32*x^3+32*x^2)/exp(x)/exp(1/5*(5*exp(x)^2+5*exp(x)*x-4*x
)/exp(x))^2,x, algorithm="maxima")

[Out]

4*x^2*e^(8/5*x*e^(-x) - 2*x - 2*e^x)

________________________________________________________________________________________

mupad [B]  time = 3.30, size = 22, normalized size = 0.85 \begin {gather*} 4\,x^2\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{\frac {8\,x\,{\mathrm {e}}^{-x}}{5}}\,{\mathrm {e}}^{-2\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-2*exp(-x)*(exp(2*x) - (4*x)/5 + x*exp(x)))*exp(-x)*(8*x^2*exp(2*x) - (exp(x)*(40*x - 40*x^2))/5 - (3
2*x^2)/5 + (32*x^3)/5),x)

[Out]

4*x^2*exp(-2*x)*exp((8*x*exp(-x))/5)*exp(-2*exp(x))

________________________________________________________________________________________

sympy [A]  time = 0.28, size = 27, normalized size = 1.04 \begin {gather*} 4 x^{2} e^{- 2 \left (x e^{x} - \frac {4 x}{5} + e^{2 x}\right ) e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-40*exp(x)**2*x**2+(-40*x**2+40*x)*exp(x)-32*x**3+32*x**2)/exp(x)/exp(1/5*(5*exp(x)**2+5*exp(x)
*x-4*x)/exp(x))**2,x)

[Out]

4*x**2*exp(-2*(x*exp(x) - 4*x/5 + exp(2*x))*exp(-x))

________________________________________________________________________________________