3.24.89 \(\int (-240 x+840 x^2-320 x^3+e^{-4+x} (360 x^2-40 x^3-40 x^4)) \, dx\)

Optimal. Leaf size=21 \[ 40 (3-x) x^2 \left (-1+\left (2+e^{-4+x}\right ) x\right ) \]

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Rubi [A]  time = 0.18, antiderivative size = 36, normalized size of antiderivative = 1.71, number of steps used = 16, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1594, 2196, 2176, 2194} \begin {gather*} -40 e^{x-4} x^4-80 x^4+120 e^{x-4} x^3+280 x^3-120 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-240*x + 840*x^2 - 320*x^3 + E^(-4 + x)*(360*x^2 - 40*x^3 - 40*x^4),x]

[Out]

-120*x^2 + 280*x^3 + 120*E^(-4 + x)*x^3 - 80*x^4 - 40*E^(-4 + x)*x^4

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-120 x^2+280 x^3-80 x^4+\int e^{-4+x} \left (360 x^2-40 x^3-40 x^4\right ) \, dx\\ &=-120 x^2+280 x^3-80 x^4+\int e^{-4+x} x^2 \left (360-40 x-40 x^2\right ) \, dx\\ &=-120 x^2+280 x^3-80 x^4+\int \left (360 e^{-4+x} x^2-40 e^{-4+x} x^3-40 e^{-4+x} x^4\right ) \, dx\\ &=-120 x^2+280 x^3-80 x^4-40 \int e^{-4+x} x^3 \, dx-40 \int e^{-4+x} x^4 \, dx+360 \int e^{-4+x} x^2 \, dx\\ &=-120 x^2+360 e^{-4+x} x^2+280 x^3-40 e^{-4+x} x^3-80 x^4-40 e^{-4+x} x^4+120 \int e^{-4+x} x^2 \, dx+160 \int e^{-4+x} x^3 \, dx-720 \int e^{-4+x} x \, dx\\ &=-720 e^{-4+x} x-120 x^2+480 e^{-4+x} x^2+280 x^3+120 e^{-4+x} x^3-80 x^4-40 e^{-4+x} x^4-240 \int e^{-4+x} x \, dx-480 \int e^{-4+x} x^2 \, dx+720 \int e^{-4+x} \, dx\\ &=720 e^{-4+x}-960 e^{-4+x} x-120 x^2+280 x^3+120 e^{-4+x} x^3-80 x^4-40 e^{-4+x} x^4+240 \int e^{-4+x} \, dx+960 \int e^{-4+x} x \, dx\\ &=960 e^{-4+x}-120 x^2+280 x^3+120 e^{-4+x} x^3-80 x^4-40 e^{-4+x} x^4-960 \int e^{-4+x} \, dx\\ &=-120 x^2+280 x^3+120 e^{-4+x} x^3-80 x^4-40 e^{-4+x} x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 26, normalized size = 1.24 \begin {gather*} -\frac {40 (-3+x) x^2 \left (e^x x+e^4 (-1+2 x)\right )}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-240*x + 840*x^2 - 320*x^3 + E^(-4 + x)*(360*x^2 - 40*x^3 - 40*x^4),x]

[Out]

(-40*(-3 + x)*x^2*(E^x*x + E^4*(-1 + 2*x)))/E^4

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fricas [A]  time = 0.79, size = 31, normalized size = 1.48 \begin {gather*} -80 \, x^{4} + 280 \, x^{3} - 120 \, x^{2} - 40 \, {\left (x^{4} - 3 \, x^{3}\right )} e^{\left (x - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-40*x^4-40*x^3+360*x^2)*exp(x-4)-320*x^3+840*x^2-240*x,x, algorithm="fricas")

[Out]

-80*x^4 + 280*x^3 - 120*x^2 - 40*(x^4 - 3*x^3)*e^(x - 4)

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giac [A]  time = 0.17, size = 31, normalized size = 1.48 \begin {gather*} -80 \, x^{4} + 280 \, x^{3} - 120 \, x^{2} - 40 \, {\left (x^{4} - 3 \, x^{3}\right )} e^{\left (x - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-40*x^4-40*x^3+360*x^2)*exp(x-4)-320*x^3+840*x^2-240*x,x, algorithm="giac")

[Out]

-80*x^4 + 280*x^3 - 120*x^2 - 40*(x^4 - 3*x^3)*e^(x - 4)

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maple [A]  time = 0.07, size = 33, normalized size = 1.57

method	result	size
risch	\(\left (-40 x^{4}+120 x^{3}\right ) {\mathrm e}^{x -4}-80 x^{4}+280 x^{3}-120 x^{2}\)	\(33\)
norman	\(-120 x^{2}+280 x^{3}-80 x^{4}-40 x^{4} {\mathrm e}^{x -4}+120 \,{\mathrm e}^{x -4} x^{3}\)	\(35\)
default	\(-4480 \,{\mathrm e}^{x -4} \left (x -4\right )-2560 \,{\mathrm e}^{x -4}-2400 \,{\mathrm e}^{x -4} \left (x -4\right )^{2}-520 \,{\mathrm e}^{x -4} \left (x -4\right )^{3}-40 \,{\mathrm e}^{x -4} \left (x -4\right )^{4}-120 x^{2}+280 x^{3}-80 x^{4}\)	\(65\)
derivativedivides	\(-960 x [phone] \,{\mathrm e}^{x -4} \left (x -4\right )-2560 \,{\mathrm e}^{x -4}-2400 \,{\mathrm e}^{x -4} \left (x -4\right )^{2}-520 \,{\mathrm e}^{x -4} \left (x -4\right )^{3}-40 \,{\mathrm e}^{x -4} \left (x -4\right )^{4}-120 \left (x -4\right )^{2}+280 x^{3}-80 x^{4}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-40*x^4-40*x^3+360*x^2)*exp(x-4)-320*x^3+840*x^2-240*x,x,method=_RETURNVERBOSE)

[Out]

(-40*x^4+120*x^3)*exp(x-4)-80*x^4+280*x^3-120*x^2

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maxima [A]  time = 0.49, size = 31, normalized size = 1.48 \begin {gather*} -80 \, x^{4} + 280 \, x^{3} - 120 \, x^{2} - 40 \, {\left (x^{4} - 3 \, x^{3}\right )} e^{\left (x - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-40*x^4-40*x^3+360*x^2)*exp(x-4)-320*x^3+840*x^2-240*x,x, algorithm="maxima")

[Out]

-80*x^4 + 280*x^3 - 120*x^2 - 40*(x^4 - 3*x^3)*e^(x - 4)

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mupad [B]  time = 1.33, size = 19, normalized size = 0.90 \begin {gather*} -40\,x^2\,\left (x-3\right )\,\left (2\,x+x\,{\mathrm {e}}^{x-4}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(840*x^2 - exp(x - 4)*(40*x^3 - 360*x^2 + 40*x^4) - 240*x - 320*x^3,x)

[Out]

-40*x^2*(x - 3)*(2*x + x*exp(x - 4) - 1)

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sympy [A]  time = 0.10, size = 29, normalized size = 1.38 \begin {gather*} - 80 x^{4} + 280 x^{3} - 120 x^{2} + \left (- 40 x^{4} + 120 x^{3}\right ) e^{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-40*x**4-40*x**3+360*x**2)*exp(x-4)-320*x**3+840*x**2-240*x,x)

[Out]

-80*x**4 + 280*x**3 - 120*x**2 + (-40*x**4 + 120*x**3)*exp(x - 4)

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