3.708 \(\int e^{8-2 x} x^3 \, dx\)

Optimal. Leaf size=52 \[ -\frac{1}{2} e^{8-2 x} x^3-\frac{3}{4} e^{8-2 x} x^2-\frac{3}{4} e^{8-2 x} x-\frac{3}{8} e^{8-2 x} \]

[Out]

(-3*E^(8 - 2*x))/8 - (3*E^(8 - 2*x)*x)/4 - (3*E^(8 - 2*x)*x^2)/4 - (E^(8 - 2*x)*x^3)/2

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Rubi [A]  time = 0.0392965, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2176, 2194} \[ -\frac{1}{2} e^{8-2 x} x^3-\frac{3}{4} e^{8-2 x} x^2-\frac{3}{4} e^{8-2 x} x-\frac{3}{8} e^{8-2 x} \]

Antiderivative was successfully verified.

[In]

Int[E^(8 - 2*x)*x^3,x]

[Out]

(-3*E^(8 - 2*x))/8 - (3*E^(8 - 2*x)*x)/4 - (3*E^(8 - 2*x)*x^2)/4 - (E^(8 - 2*x)*x^3)/2

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int e^{8-2 x} x^3 \, dx &=-\frac{1}{2} e^{8-2 x} x^3+\frac{3}{2} \int e^{8-2 x} x^2 \, dx\\ &=-\frac{3}{4} e^{8-2 x} x^2-\frac{1}{2} e^{8-2 x} x^3+\frac{3}{2} \int e^{8-2 x} x \, dx\\ &=-\frac{3}{4} e^{8-2 x} x-\frac{3}{4} e^{8-2 x} x^2-\frac{1}{2} e^{8-2 x} x^3+\frac{3}{4} \int e^{8-2 x} \, dx\\ &=-\frac{3}{8} e^{8-2 x}-\frac{3}{4} e^{8-2 x} x-\frac{3}{4} e^{8-2 x} x^2-\frac{1}{2} e^{8-2 x} x^3\\ \end{align*}

Mathematica [A]  time = 0.0097179, size = 26, normalized size = 0.5 \[ -\frac{1}{8} e^{8-2 x} \left (4 x^3+6 x^2+6 x+3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(8 - 2*x)*x^3,x]

[Out]

-(E^(8 - 2*x)*(3 + 6*x + 6*x^2 + 4*x^3))/8

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Maple [A]  time = 0.02, size = 24, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 4\,{x}^{3}+6\,{x}^{2}+6\,x+3 \right ){{\rm e}^{8-2\,x}}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(8-2*x)*x^3,x)

[Out]

-1/8*(4*x^3+6*x^2+6*x+3)*exp(8-2*x)

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Maxima [A]  time = 0.962147, size = 41, normalized size = 0.79 \begin{align*} -\frac{1}{8} \,{\left (4 \, x^{3} e^{8} + 6 \, x^{2} e^{8} + 6 \, x e^{8} + 3 \, e^{8}\right )} e^{\left (-2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(8-2*x)*x^3,x, algorithm="maxima")

[Out]

-1/8*(4*x^3*e^8 + 6*x^2*e^8 + 6*x*e^8 + 3*e^8)*e^(-2*x)

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Fricas [A]  time = 0.829967, size = 61, normalized size = 1.17 \begin{align*} -\frac{1}{8} \,{\left (4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x + 8\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(8-2*x)*x^3,x, algorithm="fricas")

[Out]

-1/8*(4*x^3 + 6*x^2 + 6*x + 3)*e^(-2*x + 8)

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Sympy [A]  time = 0.093912, size = 24, normalized size = 0.46 \begin{align*} \frac{\left (- 4 x^{3} - 6 x^{2} - 6 x - 3\right ) e^{8 - 2 x}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(8-2*x)*x**3,x)

[Out]

(-4*x**3 - 6*x**2 - 6*x - 3)*exp(8 - 2*x)/8

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Giac [A]  time = 1.27472, size = 31, normalized size = 0.6 \begin{align*} -\frac{1}{8} \,{\left (4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x + 8\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(8-2*x)*x^3,x, algorithm="giac")

[Out]

-1/8*(4*x^3 + 6*x^2 + 6*x + 3)*e^(-2*x + 8)