\(\int e^{4+8 x+4 x^2} (-24-24 x) \, dx\) [6823]

Optimal result

Integrand size = 18, antiderivative size = 22 \[ \int e^{4+8 x+4 x^2} (-24-24 x) \, dx=3 \left (-e^{(2+2 x)^2}+\frac {1}{4 \log ^2(2)}\right ) \]

[Out]

3/4/ln(2)^2-3*exp((2+2*x)^2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2268} \[ \int e^{4+8 x+4 x^2} (-24-24 x) \, dx=-3 e^{4 x^2+8 x+4} \]

[In]

Int[E^(4 + 8*x + 4*x^2)*(-24 - 24*x),x]

[Out]

-3*E^(4 + 8*x + 4*x^2)

Rule 2268

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[e*(F^(a + b*x + c*x^2)/(2
*c*Log[F])), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rubi steps \begin{align*} \text {integral}& = -3 e^{4+8 x+4 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.50 \[ \int e^{4+8 x+4 x^2} (-24-24 x) \, dx=-3 e^{4 (1+x)^2} \]

[In]

Integrate[E^(4 + 8*x + 4*x^2)*(-24 - 24*x),x]

[Out]

-3*E^(4*(1 + x)^2)

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.50

method	result	size
risch	\(-3 \,{\mathrm e}^{4 \left (1+x \right )^{2}}\)	\(11\)
gosper	\(-3 \,{\mathrm e}^{4 x^{2}+8 x +4}\)	\(14\)
norman	\(-3 \,{\mathrm e}^{4 x^{2}+8 x +4}\)	\(14\)
parallelrisch	\(-3 \,{\mathrm e}^{4 x^{2}+8 x +4}\)	\(14\)
default	\(6 i {\mathrm e}^{4} \sqrt {\pi }\, {\mathrm e}^{-4} \operatorname {erf}\left (2 i x +2 i\right )-24 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{4 x^{2}+8 x}}{8}+\frac {i \sqrt {\pi }\, {\mathrm e}^{-4} \operatorname {erf}\left (2 i x +2 i\right )}{4}\right )\)	\(53\)
parts	\(6 i \sqrt {\pi }\, \operatorname {erf}\left (2 i x +2 i\right ) x +6 i \sqrt {\pi }\, \operatorname {erf}\left (2 i x +2 i\right )-3 \sqrt {\pi }\, \left (\operatorname {erf}\left (2 i x +2 i\right ) \left (2 i x +2 i\right )+\frac {{\mathrm e}^{-\left (2 i x +2 i\right )^{2}}}{\sqrt {\pi }}\right )\)	\(69\)

[In]

int((-24*x-24)*exp(4*x^2+8*x+4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int e^{4+8 x+4 x^2} (-24-24 x) \, dx=-3 \, e^{\left (4 \, x^{2} + 8 \, x + 4\right )} \]

[In]

integrate((-24*x-24)*exp(4*x^2+8*x+4),x, algorithm="fricas")

[Out]

-3*e^(4*x^2 + 8*x + 4)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int e^{4+8 x+4 x^2} (-24-24 x) \, dx=- 3 e^{4 x^{2} + 8 x + 4} \]

[In]

integrate((-24*x-24)*exp(4*x**2+8*x+4),x)

[Out]

-3*exp(4*x**2 + 8*x + 4)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int e^{4+8 x+4 x^2} (-24-24 x) \, dx=-3 \, e^{\left (4 \, x^{2} + 8 \, x + 4\right )} \]

[In]

integrate((-24*x-24)*exp(4*x^2+8*x+4),x, algorithm="maxima")

[Out]

-3*e^(4*x^2 + 8*x + 4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int e^{4+8 x+4 x^2} (-24-24 x) \, dx=-3 \, e^{\left (4 \, x^{2} + 8 \, x + 4\right )} \]

[In]

integrate((-24*x-24)*exp(4*x^2+8*x+4),x, algorithm="giac")

[Out]

-3*e^(4*x^2 + 8*x + 4)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int e^{4+8 x+4 x^2} (-24-24 x) \, dx=-3\,{\mathrm {e}}^{8\,x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{4\,x^2} \]

[In]

int(-exp(8*x + 4*x^2 + 4)*(24*x + 24),x)

[Out]

-3*exp(8*x)*exp(4)*exp(4*x^2)