3.99.69 \(\int e^{-16 x} (675 x^2-3600 x^3) \, dx\)

Optimal. Leaf size=19 \[ -2-e^5+225 e^{-16 x} x^3+\log (4) \]

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Rubi [A]  time = 0.11, antiderivative size = 10, normalized size of antiderivative = 0.53, number of steps used = 10, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1593, 2196, 2176, 2194} \begin {gather*} 225 e^{-16 x} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(675*x^2 - 3600*x^3)/E^(16*x),x]

[Out]

(225*x^3)/E^(16*x)

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - 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} &=\int e^{-16 x} (675-3600 x) x^2 \, dx\\ &=\int \left (675 e^{-16 x} x^2-3600 e^{-16 x} x^3\right ) \, dx\\ &=675 \int e^{-16 x} x^2 \, dx-3600 \int e^{-16 x} x^3 \, dx\\ &=-\frac {675}{16} e^{-16 x} x^2+225 e^{-16 x} x^3+\frac {675}{8} \int e^{-16 x} x \, dx-675 \int e^{-16 x} x^2 \, dx\\ &=-\frac {675}{128} e^{-16 x} x+225 e^{-16 x} x^3+\frac {675}{128} \int e^{-16 x} \, dx-\frac {675}{8} \int e^{-16 x} x \, dx\\ &=-\frac {675 e^{-16 x}}{2048}+225 e^{-16 x} x^3-\frac {675}{128} \int e^{-16 x} \, dx\\ &=225 e^{-16 x} x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 0.53 \begin {gather*} 225 e^{-16 x} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(675*x^2 - 3600*x^3)/E^(16*x),x]

[Out]

(225*x^3)/E^(16*x)

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fricas [A]  time = 0.71, size = 9, normalized size = 0.47 \begin {gather*} 225 \, x^{3} e^{\left (-16 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3600*x^3+675*x^2)/exp(16*x),x, algorithm="fricas")

[Out]

225*x^3*e^(-16*x)

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giac [A]  time = 0.13, size = 9, normalized size = 0.47 \begin {gather*} 225 \, x^{3} e^{\left (-16 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3600*x^3+675*x^2)/exp(16*x),x, algorithm="giac")

[Out]

225*x^3*e^(-16*x)

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maple [A]  time = 0.04, size = 10, normalized size = 0.53

method	result	size
risch	\(225 x^{3} {\mathrm e}^{-16 x}\)	\(10\)
gosper	\(225 x^{3} {\mathrm e}^{-16 x}\)	\(12\)
derivativedivides	\(225 x^{3} {\mathrm e}^{-16 x}\)	\(12\)
default	\(225 x^{3} {\mathrm e}^{-16 x}\)	\(12\)
norman	\(225 x^{3} {\mathrm e}^{-16 x}\)	\(12\)
meijerg	\(\frac {225 \left (16384 x^{3}+3072 x^{2}+384 x +24\right ) {\mathrm e}^{-16 x}}{16384}-\frac {225 \left (768 x^{2}+96 x +6\right ) {\mathrm e}^{-16 x}}{4096}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3600*x^3+675*x^2)/exp(16*x),x,method=_RETURNVERBOSE)

[Out]

225*x^3*exp(-16*x)

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maxima [A]  time = 0.36, size = 38, normalized size = 2.00 \begin {gather*} \frac {225}{2048} \, {\left (2048 \, x^{3} + 384 \, x^{2} + 48 \, x + 3\right )} e^{\left (-16 \, x\right )} - \frac {675}{2048} \, {\left (128 \, x^{2} + 16 \, x + 1\right )} e^{\left (-16 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3600*x^3+675*x^2)/exp(16*x),x, algorithm="maxima")

[Out]

225/2048*(2048*x^3 + 384*x^2 + 48*x + 3)*e^(-16*x) - 675/2048*(128*x^2 + 16*x + 1)*e^(-16*x)

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mupad [B]  time = 5.58, size = 9, normalized size = 0.47 \begin {gather*} 225\,x^3\,{\mathrm {e}}^{-16\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-16*x)*(675*x^2 - 3600*x^3),x)

[Out]

225*x^3*exp(-16*x)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.42 \begin {gather*} 225 x^{3} e^{- 16 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3600*x**3+675*x**2)/exp(16*x),x)

[Out]

225*x**3*exp(-16*x)

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