3.39.93 \(\int e^{-3-x} (-5+5 x+e^{3+x} (20736+13828 x+2592 x^2+192 x^3+5 x^4)) \, dx\)

Optimal. Leaf size=22 \[ x^2+x \left (-5 e^{-3-x}+x+(12+x)^4\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 32, normalized size of antiderivative = 1.45, number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6742, 2194, 2176} \begin {gather*} x^5+48 x^4+864 x^3+6914 x^2-5 e^{-x-3} x+20736 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-3 - x)*(-5 + 5*x + E^(3 + x)*(20736 + 13828*x + 2592*x^2 + 192*x^3 + 5*x^4)),x]

[Out]

20736*x - 5*E^(-3 - x)*x + 6914*x^2 + 864*x^3 + 48*x^4 + x^5

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]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (20736-5 e^{-3-x}+13828 x+5 e^{-3-x} x+2592 x^2+192 x^3+5 x^4\right ) \, dx\\ &=20736 x+6914 x^2+864 x^3+48 x^4+x^5-5 \int e^{-3-x} \, dx+5 \int e^{-3-x} x \, dx\\ &=5 e^{-3-x}+20736 x-5 e^{-3-x} x+6914 x^2+864 x^3+48 x^4+x^5+5 \int e^{-3-x} \, dx\\ &=20736 x-5 e^{-3-x} x+6914 x^2+864 x^3+48 x^4+x^5\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 29, normalized size = 1.32 \begin {gather*} x \left (20736-5 e^{-3-x}+6914 x+864 x^2+48 x^3+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-3 - x)*(-5 + 5*x + E^(3 + x)*(20736 + 13828*x + 2592*x^2 + 192*x^3 + 5*x^4)),x]

[Out]

x*(20736 - 5*E^(-3 - x) + 6914*x + 864*x^2 + 48*x^3 + x^4)

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fricas [A]  time = 0.58, size = 38, normalized size = 1.73 \begin {gather*} {\left ({\left (x^{5} + 48 \, x^{4} + 864 \, x^{3} + 6914 \, x^{2} + 20736 \, x\right )} e^{\left (x + 3\right )} - 5 \, x\right )} e^{\left (-x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4+192*x^3+2592*x^2+13828*x+20736)*exp(3+x)+5*x-5)/exp(3+x),x, algorithm="fricas")

[Out]

((x^5 + 48*x^4 + 864*x^3 + 6914*x^2 + 20736*x)*e^(x + 3) - 5*x)*e^(-x - 3)

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giac [B]  time = 0.13, size = 43, normalized size = 1.95 \begin {gather*} {\left (x^{5} e^{3} + 48 \, x^{4} e^{3} + 864 \, x^{3} e^{3} + 6914 \, x^{2} e^{3} + 20736 \, x e^{3} - 5 \, x e^{\left (-x\right )}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4+192*x^3+2592*x^2+13828*x+20736)*exp(3+x)+5*x-5)/exp(3+x),x, algorithm="giac")

[Out]

(x^5*e^3 + 48*x^4*e^3 + 864*x^3*e^3 + 6914*x^2*e^3 + 20736*x*e^3 - 5*x*e^(-x))*e^(-3)

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maple [A]  time = 0.03, size = 32, normalized size = 1.45

method	result	size
risch	\(x^{5}+48 x^{4}+864 x^{3}+6914 x^{2}+20736 x -5 x \,{\mathrm e}^{-3-x}\)	\(32\)
derivativedivides	\(\left (3+x \right )^{5}+33 \left (3+x \right )^{4}+378 \left (3+x \right )^{3}+1460 \left (3+x \right )^{2}-6597-2199 x +15 \,{\mathrm e}^{-3-x}-5 \,{\mathrm e}^{-3-x} \left (3+x \right )\)	\(51\)
default	\(\left (3+x \right )^{5}+33 \left (3+x \right )^{4}+378 \left (3+x \right )^{3}+1460 \left (3+x \right )^{2}-6597-2199 x +15 \,{\mathrm e}^{-3-x}-5 \,{\mathrm e}^{-3-x} \left (3+x \right )\)	\(51\)
norman	\(\left (x^{5} {\mathrm e}^{3+x}-5 x +6914 x^{2} {\mathrm e}^{3+x}+864 x^{3} {\mathrm e}^{3+x}+48 x^{4} {\mathrm e}^{3+x}+20736 \,{\mathrm e}^{3+x} x \right ) {\mathrm e}^{-3-x}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^4+192*x^3+2592*x^2+13828*x+20736)*exp(3+x)+5*x-5)/exp(3+x),x,method=_RETURNVERBOSE)

[Out]

x^5+48*x^4+864*x^3+6914*x^2+20736*x-5*x*exp(-3-x)

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maxima [A]  time = 0.35, size = 41, normalized size = 1.86 \begin {gather*} x^{5} + 48 \, x^{4} + 864 \, x^{3} + 6914 \, x^{2} - 5 \, {\left (x + 1\right )} e^{\left (-x - 3\right )} + 20736 \, x + 5 \, e^{\left (-x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4+192*x^3+2592*x^2+13828*x+20736)*exp(3+x)+5*x-5)/exp(3+x),x, algorithm="maxima")

[Out]

x^5 + 48*x^4 + 864*x^3 + 6914*x^2 - 5*(x + 1)*e^(-x - 3) + 20736*x + 5*e^(-x - 3)

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mupad [B]  time = 2.19, size = 28, normalized size = 1.27 \begin {gather*} x\,\left (6914\,x-5\,{\mathrm {e}}^{-x-3}+864\,x^2+48\,x^3+x^4+20736\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(- x - 3)*(5*x + exp(x + 3)*(13828*x + 2592*x^2 + 192*x^3 + 5*x^4 + 20736) - 5),x)

[Out]

x*(6914*x - 5*exp(- x - 3) + 864*x^2 + 48*x^3 + x^4 + 20736)

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sympy [A]  time = 0.12, size = 31, normalized size = 1.41 \begin {gather*} x^{5} + 48 x^{4} + 864 x^{3} + 6914 x^{2} - 5 x e^{- x - 3} + 20736 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**4+192*x**3+2592*x**2+13828*x+20736)*exp(3+x)+5*x-5)/exp(3+x),x)

[Out]

x**5 + 48*x**4 + 864*x**3 + 6914*x**2 - 5*x*exp(-x - 3) + 20736*x

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