\(\int e^{2 e^{6+x} x+e^6 (4 x^2-4 x^3)} (e^{6+x} (2+2 x)+e^6 (8 x-12 x^2)) \, dx\) [199]

Optimal result

Integrand size = 52, antiderivative size = 22 \[ \int e^{2 e^{6+x} x+e^6 \left (4 x^2-4 x^3\right )} \left (e^{6+x} (2+2 x)+e^6 \left (8 x-12 x^2\right )\right ) \, dx=e^{4 e^6 x \left (\frac {e^x}{2}+x-x^2\right )} \]

[Out]

exp(4*exp(6)*x*(1/2*exp(x)-x^2+x))

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6838} \[ \int e^{2 e^{6+x} x+e^6 \left (4 x^2-4 x^3\right )} \left (e^{6+x} (2+2 x)+e^6 \left (8 x-12 x^2\right )\right ) \, dx=e^{4 e^6 \left (x^2-x^3\right )+2 e^{x+6} x} \]

[In]

Int[E^(2*E^(6 + x)*x + E^6*(4*x^2 - 4*x^3))*(E^(6 + x)*(2 + 2*x) + E^6*(8*x - 12*x^2)),x]

[Out]

E^(2*E^(6 + x)*x + 4*E^6*(x^2 - x^3))

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = e^{2 e^{6+x} x+4 e^6 \left (x^2-x^3\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int e^{2 e^{6+x} x+e^6 \left (4 x^2-4 x^3\right )} \left (e^{6+x} (2+2 x)+e^6 \left (8 x-12 x^2\right )\right ) \, dx=e^{2 e^6 x \left (e^x-2 (-1+x) x\right )} \]

[In]

Integrate[E^(2*E^(6 + x)*x + E^6*(4*x^2 - 4*x^3))*(E^(6 + x)*(2 + 2*x) + E^6*(8*x - 12*x^2)),x]

[Out]

E^(2*E^6*x*(E^x - 2*(-1 + x)*x))

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82

method	result	size
parallelrisch	\({\mathrm e}^{2 \,{\mathrm e}^{6} x \left (-2 x^{2}+{\mathrm e}^{x}+2 x \right )}\)	\(18\)
norman	\({\mathrm e}^{2 x \,{\mathrm e}^{6} {\mathrm e}^{x}+\left (-4 x^{3}+4 x^{2}\right ) {\mathrm e}^{6}}\)	\(24\)
risch	\({\mathrm e}^{-2 x \left (2 x^{2} {\mathrm e}^{6}-2 x \,{\mathrm e}^{6}-{\mathrm e}^{6+x}\right )}\)	\(24\)

[In]

int(((2+2*x)*exp(6)*exp(x)+(-12*x^2+8*x)*exp(6))*exp(2*x*exp(6)*exp(x)+(-4*x^3+4*x^2)*exp(6)),x,method=_RETURN
VERBOSE)

[Out]

exp(2*exp(6)*x*(-2*x^2+exp(x)+2*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int e^{2 e^{6+x} x+e^6 \left (4 x^2-4 x^3\right )} \left (e^{6+x} (2+2 x)+e^6 \left (8 x-12 x^2\right )\right ) \, dx=e^{\left (-4 \, {\left (x^{3} - x^{2}\right )} e^{6} + 2 \, x e^{\left (x + 6\right )}\right )} \]

[In]

integrate(((2+2*x)*exp(6)*exp(x)+(-12*x^2+8*x)*exp(6))*exp(2*x*exp(6)*exp(x)+(-4*x^3+4*x^2)*exp(6)),x, algorit
hm="fricas")

[Out]

e^(-4*(x^3 - x^2)*e^6 + 2*x*e^(x + 6))

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int e^{2 e^{6+x} x+e^6 \left (4 x^2-4 x^3\right )} \left (e^{6+x} (2+2 x)+e^6 \left (8 x-12 x^2\right )\right ) \, dx=e^{2 x e^{6} e^{x} + \left (- 4 x^{3} + 4 x^{2}\right ) e^{6}} \]

[In]

integrate(((2+2*x)*exp(6)*exp(x)+(-12*x**2+8*x)*exp(6))*exp(2*x*exp(6)*exp(x)+(-4*x**3+4*x**2)*exp(6)),x)

[Out]

exp(2*x*exp(6)*exp(x) + (-4*x**3 + 4*x**2)*exp(6))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int e^{2 e^{6+x} x+e^6 \left (4 x^2-4 x^3\right )} \left (e^{6+x} (2+2 x)+e^6 \left (8 x-12 x^2\right )\right ) \, dx=e^{\left (-4 \, x^{3} e^{6} + 4 \, x^{2} e^{6} + 2 \, x e^{\left (x + 6\right )}\right )} \]

[In]

integrate(((2+2*x)*exp(6)*exp(x)+(-12*x^2+8*x)*exp(6))*exp(2*x*exp(6)*exp(x)+(-4*x^3+4*x^2)*exp(6)),x, algorit
hm="maxima")

[Out]

e^(-4*x^3*e^6 + 4*x^2*e^6 + 2*x*e^(x + 6))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int e^{2 e^{6+x} x+e^6 \left (4 x^2-4 x^3\right )} \left (e^{6+x} (2+2 x)+e^6 \left (8 x-12 x^2\right )\right ) \, dx=e^{\left (-4 \, x^{3} e^{6} + 4 \, x^{2} e^{6} + 2 \, x e^{\left (x + 6\right )}\right )} \]

[In]

integrate(((2+2*x)*exp(6)*exp(x)+(-12*x^2+8*x)*exp(6))*exp(2*x*exp(6)*exp(x)+(-4*x^3+4*x^2)*exp(6)),x, algorit
hm="giac")

[Out]

e^(-4*x^3*e^6 + 4*x^2*e^6 + 2*x*e^(x + 6))

Mupad [B] (verification not implemented)

Time = 7.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int e^{2 e^{6+x} x+e^6 \left (4 x^2-4 x^3\right )} \left (e^{6+x} (2+2 x)+e^6 \left (8 x-12 x^2\right )\right ) \, dx={\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^6}\,{\mathrm {e}}^{-4\,x^3\,{\mathrm {e}}^6}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^6\,{\mathrm {e}}^x} \]

[In]

int(exp(exp(6)*(4*x^2 - 4*x^3) + 2*x*exp(6)*exp(x))*(exp(6)*(8*x - 12*x^2) + exp(6)*exp(x)*(2*x + 2)),x)

[Out]

exp(4*x^2*exp(6))*exp(-4*x^3*exp(6))*exp(2*x*exp(6)*exp(x))