3.101.60 \(\int e^{\frac {1}{2} e^2 (8 x-x^2)} (3+e^2 (12 x-3 x^2)) \, dx\)

Optimal. Leaf size=20 \[ -20+3 e^{\frac {1}{2} e^2 (8-x) x} x \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.80, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2288} \begin {gather*} \frac {3 e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (4 x-x^2\right )}{4-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((E^2*(8*x - x^2))/2)*(3 + E^2*(12*x - 3*x^2)),x]

[Out]

(3*E^((E^2*(8*x - x^2))/2)*(4*x - x^2))/(4 - x)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {3 e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (4 x-x^2\right )}{4-x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 16, normalized size = 0.80 \begin {gather*} 3 e^{-\frac {1}{2} e^2 (-8+x) x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((E^2*(8*x - x^2))/2)*(3 + E^2*(12*x - 3*x^2)),x]

[Out]

(3*x)/E^((E^2*(-8 + x)*x)/2)

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 15, normalized size = 0.75 \begin {gather*} 3 \, x e^{\left (-\frac {1}{2} \, {\left (x^{2} - 8 \, x\right )} e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+12*x)*exp(2)+3)*exp(1/2*(-x^2+8*x)*exp(2)),x, algorithm="fricas")

[Out]

3*x*e^(-1/2*(x^2 - 8*x)*e^2)

________________________________________________________________________________________

giac [A]  time = 0.24, size = 17, normalized size = 0.85 \begin {gather*} 3 \, x e^{\left (-\frac {1}{2} \, x^{2} e^{2} + 4 \, x e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+12*x)*exp(2)+3)*exp(1/2*(-x^2+8*x)*exp(2)),x, algorithm="giac")

[Out]

3*x*e^(-1/2*x^2*e^2 + 4*x*e^2)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 13, normalized size = 0.65

method	result	size
gosper	\(3 \,{\mathrm e}^{-\frac {x \left (-8+x \right ) {\mathrm e}^{2}}{2}} x\)	\(13\)
risch	\(3 \,{\mathrm e}^{-\frac {x \left (-8+x \right ) {\mathrm e}^{2}}{2}} x\)	\(13\)
norman	\(3 x \,{\mathrm e}^{\frac {\left (-x^{2}+8 x \right ) {\mathrm e}^{2}}{2}}\)	\(18\)
default	\(\frac {3 \sqrt {\pi }\, {\mathrm e}^{8 \,{\mathrm e}^{2}} \sqrt {2}\, {\mathrm e}^{-1} \erf \left (\frac {\sqrt {2}\, {\mathrm e} x}{2}-2 \,{\mathrm e}^{2} \sqrt {2}\, {\mathrm e}^{-1}\right )}{2}+3 \,{\mathrm e}^{-2} x \,{\mathrm e}^{-\frac {x^{2} {\mathrm e}^{2}}{2}+4 \,{\mathrm e}^{2} x +2}-\frac {3 \,{\mathrm e}^{-2} \sqrt {\pi }\, {\mathrm e}^{2+8 \,{\mathrm e}^{2}} \sqrt {2}\, {\mathrm e}^{-1} \erf \left (\frac {\sqrt {2}\, {\mathrm e} x}{2}-2 \,{\mathrm e}^{2} \sqrt {2}\, {\mathrm e}^{-1}\right )}{2}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^2+12*x)*exp(2)+3)*exp(1/2*(-x^2+8*x)*exp(2)),x,method=_RETURNVERBOSE)

[Out]

3*exp(-1/2*x*(-8+x)*exp(2))*x

________________________________________________________________________________________

maxima [C]  time = 0.52, size = 311, normalized size = 15.55 \begin {gather*} \frac {3}{2} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} x e - 2 \, \sqrt {2} e\right ) e^{\left (8 \, e^{2} - 1\right )} + \frac {24 \, \sqrt {\frac {1}{2}} {\left (\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\pi } {\left (x e^{2} - 4 \, e^{2}\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (x e^{2} - 4 \, e^{2}\right )}^{2}} e^{\left (-1\right )}\right ) - 1\right )} e^{3}}{\sqrt {{\left (x e^{2} - 4 \, e^{2}\right )}^{2}} \left (-e^{2}\right )^{\frac {3}{2}}} - \frac {\sqrt {\frac {1}{2}} e^{\left (-\frac {1}{2} \, {\left (x e^{2} - 4 \, e^{2}\right )}^{2} e^{\left (-2\right )} + 2\right )}}{\left (-e^{2}\right )^{\frac {3}{2}}}\right )} e^{\left (8 \, e^{2} + 2\right )}}{\sqrt {-e^{2}}} - \frac {6 \, \sqrt {\frac {1}{2}} {\left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} {\left (x e^{2} - 4 \, e^{2}\right )}^{3} e^{3} \Gamma \left (\frac {3}{2}, \frac {1}{2} \, {\left (x e^{2} - 4 \, e^{2}\right )}^{2} e^{\left (-2\right )}\right )}{{\left ({\left (x e^{2} - 4 \, e^{2}\right )}^{2}\right )}^{\frac {3}{2}} \left (-e^{2}\right )^{\frac {5}{2}}} - \frac {8 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\pi } {\left (x e^{2} - 4 \, e^{2}\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (x e^{2} - 4 \, e^{2}\right )}^{2}} e^{\left (-1\right )}\right ) - 1\right )} e^{5}}{\sqrt {{\left (x e^{2} - 4 \, e^{2}\right )}^{2}} \left (-e^{2}\right )^{\frac {5}{2}}} + \frac {8 \, \sqrt {\frac {1}{2}} e^{\left (-\frac {1}{2} \, {\left (x e^{2} - 4 \, e^{2}\right )}^{2} e^{\left (-2\right )} + 4\right )}}{\left (-e^{2}\right )^{\frac {5}{2}}}\right )} e^{\left (8 \, e^{2} + 2\right )}}{\sqrt {-e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+12*x)*exp(2)+3)*exp(1/2*(-x^2+8*x)*exp(2)),x, algorithm="maxima")

[Out]

3/2*sqrt(2)*sqrt(pi)*erf(1/2*sqrt(2)*x*e - 2*sqrt(2)*e)*e^(8*e^2 - 1) + 24*sqrt(1/2)*(2*sqrt(2)*sqrt(1/2)*sqrt
(pi)*(x*e^2 - 4*e^2)*(erf(sqrt(1/2)*sqrt((x*e^2 - 4*e^2)^2)*e^(-1)) - 1)*e^3/(sqrt((x*e^2 - 4*e^2)^2)*(-e^2)^(
3/2)) - sqrt(1/2)*e^(-1/2*(x*e^2 - 4*e^2)^2*e^(-2) + 2)/(-e^2)^(3/2))*e^(8*e^2 + 2)/sqrt(-e^2) - 6*sqrt(1/2)*(
sqrt(2)*sqrt(1/2)*(x*e^2 - 4*e^2)^3*e^3*gamma(3/2, 1/2*(x*e^2 - 4*e^2)^2*e^(-2))/(((x*e^2 - 4*e^2)^2)^(3/2)*(-
e^2)^(5/2)) - 8*sqrt(2)*sqrt(1/2)*sqrt(pi)*(x*e^2 - 4*e^2)*(erf(sqrt(1/2)*sqrt((x*e^2 - 4*e^2)^2)*e^(-1)) - 1)
*e^5/(sqrt((x*e^2 - 4*e^2)^2)*(-e^2)^(5/2)) + 8*sqrt(1/2)*e^(-1/2*(x*e^2 - 4*e^2)^2*e^(-2) + 4)/(-e^2)^(5/2))*
e^(8*e^2 + 2)/sqrt(-e^2)

________________________________________________________________________________________

mupad [B]  time = 0.16, size = 17, normalized size = 0.85 \begin {gather*} 3\,x\,{\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^2}{2}}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((exp(2)*(8*x - x^2))/2)*(exp(2)*(12*x - 3*x^2) + 3),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 15, normalized size = 0.75 \begin {gather*} 3 x e^{\left (- \frac {x^{2}}{2} + 4 x\right ) e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**2+12*x)*exp(2)+3)*exp(1/2*(-x**2+8*x)*exp(2)),x)

[Out]

3*x*exp((-x**2/2 + 4*x)*exp(2))

________________________________________________________________________________________