3.53.25 \(\int 65536 e^{-3-14 x} (3 x^2-14 x^3) \, dx\)

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

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 12, normalized size of antiderivative = 0.63, number of steps used = 11, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {12, 1593, 2196, 2176, 2194} \begin {gather*} 65536 e^{-14 x-3} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[65536*E^(-3 - 14*x)*(3*x^2 - 14*x^3),x]

[Out]

65536*E^(-3 - 14*x)*x^3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=65536 \int e^{-3-14 x} \left (3 x^2-14 x^3\right ) \, dx\\ &=65536 \int e^{-3-14 x} (3-14 x) x^2 \, dx\\ &=65536 \int \left (3 e^{-3-14 x} x^2-14 e^{-3-14 x} x^3\right ) \, dx\\ &=196608 \int e^{-3-14 x} x^2 \, dx-917504 \int e^{-3-14 x} x^3 \, dx\\ &=-\frac {98304}{7} e^{-3-14 x} x^2+65536 e^{-3-14 x} x^3+\frac {196608}{7} \int e^{-3-14 x} x \, dx-196608 \int e^{-3-14 x} x^2 \, dx\\ &=-\frac {98304}{49} e^{-3-14 x} x+65536 e^{-3-14 x} x^3+\frac {98304}{49} \int e^{-3-14 x} \, dx-\frac {196608}{7} \int e^{-3-14 x} x \, dx\\ &=-\frac {49152}{343} e^{-3-14 x}+65536 e^{-3-14 x} x^3-\frac {98304}{49} \int e^{-3-14 x} \, dx\\ &=65536 e^{-3-14 x} x^3\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 12, normalized size = 0.63 \begin {gather*} 65536 e^{-3-14 x} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[65536*E^(-3 - 14*x)*(3*x^2 - 14*x^3),x]

[Out]

65536*E^(-3 - 14*x)*x^3

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 14, normalized size = 0.74 \begin {gather*} x^{3} e^{\left (-14 \, x + 16 \, \log \relax (2) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-14*x^3+3*x^2)*exp(x)/exp(-16*log(2)+15*x+3),x, algorithm="fricas")

[Out]

x^3*e^(-14*x + 16*log(2) - 3)

________________________________________________________________________________________

giac [A]  time = 0.22, size = 14, normalized size = 0.74 \begin {gather*} x^{3} e^{\left (-14 \, x + 16 \, \log \relax (2) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-14*x^3+3*x^2)*exp(x)/exp(-16*log(2)+15*x+3),x, algorithm="giac")

[Out]

x^3*e^(-14*x + 16*log(2) - 3)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 12, normalized size = 0.63

method	result	size
risch	\(65536 x^{3} {\mathrm e}^{-14 x -3}\)	\(12\)
gosper	\(65536 x^{3} {\mathrm e}^{-15 x -3} {\mathrm e}^{x}\)	\(19\)
default	\(196608 \,{\mathrm e}^{-3} \left (-\frac {{\mathrm e}^{-14 x} x^{2}}{14}-\frac {x \,{\mathrm e}^{-14 x}}{98}-\frac {{\mathrm e}^{-14 x}}{1372}\right )-917504 \,{\mathrm e}^{-3} \left (-\frac {{\mathrm e}^{-14 x} x^{3}}{14}-\frac {3 \,{\mathrm e}^{-14 x} x^{2}}{196}-\frac {3 x \,{\mathrm e}^{-14 x}}{1372}-\frac {3 \,{\mathrm e}^{-14 x}}{19208}\right )\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-14*x^3+3*x^2)*exp(x)/exp(-16*ln(2)+15*x+3),x,method=_RETURNVERBOSE)

[Out]

65536*x^3*exp(-14*x-3)

________________________________________________________________________________________

maxima [B]  time = 0.35, size = 42, normalized size = 2.21 \begin {gather*} \frac {16384}{343} \, {\left (1372 \, x^{3} + 294 \, x^{2} + 42 \, x + 3\right )} e^{\left (-14 \, x - 3\right )} - \frac {49152}{343} \, {\left (98 \, x^{2} + 14 \, x + 1\right )} e^{\left (-14 \, x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-14*x^3+3*x^2)*exp(x)/exp(-16*log(2)+15*x+3),x, algorithm="maxima")

[Out]

16384/343*(1372*x^3 + 294*x^2 + 42*x + 3)*e^(-14*x - 3) - 49152/343*(98*x^2 + 14*x + 1)*e^(-14*x - 3)

________________________________________________________________________________________

mupad [B]  time = 3.29, size = 11, normalized size = 0.58 \begin {gather*} 65536\,x^3\,{\mathrm {e}}^{-14\,x}\,{\mathrm {e}}^{-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(16*log(2) - 15*x - 3)*exp(x)*(3*x^2 - 14*x^3),x)

[Out]

65536*x^3*exp(-14*x)*exp(-3)

________________________________________________________________________________________

sympy [A]  time = 0.22, size = 12, normalized size = 0.63 \begin {gather*} \frac {65536 x^{3} e^{- 14 x}}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-14*x**3+3*x**2)*exp(x)/exp(-16*ln(2)+15*x+3),x)

[Out]

65536*x**3*exp(-3)*exp(-14*x)

________________________________________________________________________________________