\(\int 65536 e^{-3-14 x} (3 x^2-14 x^3) \, dx\) [2225]

Optimal result

Integrand size = 20, antiderivative size = 19 \[ \int 65536 e^{-3-14 x} \left (3 x^2-14 x^3\right ) \, dx=e^{-3+2 x-16 (x-\log (2))} x^3 \] Output:

x^3/exp(-16*ln(2)+15*x+3)*exp(x)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int 65536 e^{-3-14 x} \left (3 x^2-14 x^3\right ) \, dx=65536 e^{-3-14 x} x^3 \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {27, 2027, 2626, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 27

\(\displaystyle 65536 \int e^{-14 x-3} \left (3 x^2-14 x^3\right )dx\)

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 2626

\(\displaystyle 65536 \int \left (3 e^{-14 x-3} x^2-14 e^{-14 x-3} x^3\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 65536 e^{-14 x-3} x^3\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2626

Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr 
eeQ[F, x] && PolynomialQ[Px, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63

method	result	size
risch	\(65536 x^{3} {\mathrm e}^{-14 x -3}\)	\(12\)
gosper	\(65536 x^{3} {\mathrm e}^{-3-15 x} {\mathrm e}^{x}\)	\(19\)
parallelrisch	\(65536 x^{3} {\mathrm e}^{-3-15 x} {\mathrm e}^{x}\)	\(19\)
orering	\(-\frac {65536 x \left (-14 x^{3}+3 x^{2}\right ) {\mathrm e}^{x} {\mathrm e}^{-3-15 x}}{14 x -3}\)	\(36\)
default	\(196608 \,{\mathrm e}^{-3} \left (-\frac {{\mathrm e}^{-14 x} x^{2}}{14}-\frac {x \,{\mathrm e}^{-14 x}}{98}-\frac {{\mathrm e}^{-14 x}}{1372}\right )-917504 \,{\mathrm e}^{-3} \left (-\frac {{\mathrm e}^{-14 x} x^{3}}{14}-\frac {3 \,{\mathrm e}^{-14 x} x^{2}}{196}-\frac {3 x \,{\mathrm e}^{-14 x}}{1372}-\frac {3 \,{\mathrm e}^{-14 x}}{19208}\right )\)	\(69\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int 65536 e^{-3-14 x} \left (3 x^2-14 x^3\right ) \, dx=x^{3} e^{\left (-14 \, x + 16 \, \log \left (2\right ) - 3\right )} \] Input:

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

Output:

x^3*e^(-14*x + 16*log(2) - 3)
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int 65536 e^{-3-14 x} \left (3 x^2-14 x^3\right ) \, dx=\frac {65536 x^{3} e^{- 14 x}}{e^{3}} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (14) = 28\).

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int 65536 e^{-3-14 x} \left (3 x^2-14 x^3\right ) \, dx=\frac {16384}{343} \, {\left (1372 \, x^{3} + 294 \, x^{2} + 42 \, x + 3\right )} e^{\left (-14 \, x - 3\right )} - \frac {49152}{343} \, {\left (98 \, x^{2} + 14 \, x + 1\right )} e^{\left (-14 \, x - 3\right )} \] Input:

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

Output:

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)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int 65536 e^{-3-14 x} \left (3 x^2-14 x^3\right ) \, dx=x^{3} e^{\left (-14 \, x + 16 \, \log \left (2\right ) - 3\right )} \] Input:

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

Output:

x^3*e^(-14*x + 16*log(2) - 3)
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int 65536 e^{-3-14 x} \left (3 x^2-14 x^3\right ) \, dx=65536\,x^3\,{\mathrm {e}}^{-14\,x}\,{\mathrm {e}}^{-3} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int 65536 e^{-3-14 x} \left (3 x^2-14 x^3\right ) \, dx=\frac {65536 x^{3}}{e^{14 x} e^{3}} \] Input:

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

Output:

(65536*x**3)/(e**(14*x)*e**3)