3.6.10 \(\int e^{-x} (4 e^x x+e^{-3+x+x^2} (8 x^3+4 x^5)) \, dx\)

Optimal. Leaf size=26 \[ 2 e^{-x} x^2 \left (e^x+e^{-3+x+x^2} x^2\right ) \]

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Rubi [A]  time = 0.17, antiderivative size = 18, normalized size of antiderivative = 0.69, number of steps used = 9, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6688, 2226, 2212, 2209} \begin {gather*} 2 x^2+2 e^{x^2-3} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^x*x + E^(-3 + x + x^2)*(8*x^3 + 4*x^5))/E^x,x]

[Out]

2*x^2 + 2*E^(-3 + x^2)*x^4

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 18, normalized size = 0.69 \begin {gather*} 2 x^2+2 e^{-3+x^2} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^x*x + E^(-3 + x + x^2)*(8*x^3 + 4*x^5))/E^x,x]

[Out]

2*x^2 + 2*E^(-3 + x^2)*x^4

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fricas [A]  time = 0.87, size = 24, normalized size = 0.92 \begin {gather*} 2 \, {\left (x^{4} e^{\left (x^{2} + x - 3\right )} + x^{2} e^{x}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+8*x^3)*exp(x^2+x-3)+4*exp(x)*x)/exp(x),x, algorithm="fricas")

[Out]

2*(x^4*e^(x^2 + x - 3) + x^2*e^x)*e^(-x)

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giac [A]  time = 0.60, size = 37, normalized size = 1.42 \begin {gather*} 2 \, {\left (x^{2} e^{3} + {\left (x^{4} - 2 \, x^{2} + 2\right )} e^{\left (x^{2}\right )} + 2 \, {\left (x^{2} - 1\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+8*x^3)*exp(x^2+x-3)+4*exp(x)*x)/exp(x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 18, normalized size = 0.69

method	result	size
risch	\(2 x^{2}+2 x^{4} {\mathrm e}^{x^{2}-3}\)	\(18\)
norman	\(\left (2 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x^{2}+x -3} x^{4}\right ) {\mathrm e}^{-x}\)	\(26\)
default	\(2 x^{2}+8 \,{\mathrm e}^{-3} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+4 \,{\mathrm e}^{-3} \left (\frac {x^{4} {\mathrm e}^{x^{2}}}{2}-x^{2} {\mathrm e}^{x^{2}}+{\mathrm e}^{x^{2}}\right )\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^5+8*x^3)*exp(x^2+x-3)+4*exp(x)*x)/exp(x),x,method=_RETURNVERBOSE)

[Out]

2*x^2+2*x^4*exp(x^2-3)

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maxima [A]  time = 0.37, size = 37, normalized size = 1.42 \begin {gather*} 2 \, x^{2} + 2 \, {\left (x^{4} - 2 \, x^{2} + 2\right )} e^{\left (x^{2} - 3\right )} + 4 \, {\left (x^{2} - 1\right )} e^{\left (x^{2} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+8*x^3)*exp(x^2+x-3)+4*exp(x)*x)/exp(x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.55, size = 17, normalized size = 0.65 \begin {gather*} 2\,x^2+2\,x^4\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^2 + 2*x^4*exp(x^2)*exp(-3)

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sympy [A]  time = 0.18, size = 20, normalized size = 0.77 \begin {gather*} 2 x^{4} e^{- x} e^{x^{2} + x - 3} + 2 x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**5+8*x**3)*exp(x**2+x-3)+4*exp(x)*x)/exp(x),x)

[Out]

2*x**4*exp(-x)*exp(x**2 + x - 3) + 2*x**2

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