\(\int (e^3+e^{x+4 e^x x} (1+e^x (4+4 x))) \, dx\) [6462]

Optimal result

Integrand size = 26, antiderivative size = 18 \[ \int \left (e^3+e^{x+4 e^x x} \left (1+e^x (4+4 x)\right )\right ) \, dx=e^{x+4 e^x x}+e^3 (5+x) \]

[Out]

exp(4*exp(x)*x+x)+exp(3)*(5+x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6838} \[ \int \left (e^3+e^{x+4 e^x x} \left (1+e^x (4+4 x)\right )\right ) \, dx=e^3 x+e^{4 e^x x+x} \]

[In]

Int[E^3 + E^(x + 4*E^x*x)*(1 + E^x*(4 + 4*x)),x]

[Out]

E^(x + 4*E^x*x) + E^3*x

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^3 x+\int e^{x+4 e^x x} \left (1+e^x (4+4 x)\right ) \, dx \\ & = e^{x+4 e^x x}+e^3 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \left (e^3+e^{x+4 e^x x} \left (1+e^x (4+4 x)\right )\right ) \, dx=e^{x+4 e^x x}+e^3 x \]

[In]

Integrate[E^3 + E^(x + 4*E^x*x)*(1 + E^x*(4 + 4*x)),x]

[Out]

E^(x + 4*E^x*x) + E^3*x

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78

method	result	size
default	\({\mathrm e}^{4 \,{\mathrm e}^{x} x +x}+x \,{\mathrm e}^{3}\)	\(14\)
norman	\({\mathrm e}^{4 \,{\mathrm e}^{x} x +x}+x \,{\mathrm e}^{3}\)	\(14\)
risch	\({\mathrm e}^{x \left (4 \,{\mathrm e}^{x}+1\right )}+x \,{\mathrm e}^{3}\)	\(15\)
parallelrisch	\({\mathrm e}^{x \left (4 \,{\mathrm e}^{x}+1\right )}+x \,{\mathrm e}^{3}\)	\(15\)

[In]

int(((4+4*x)*exp(x)+1)*exp(4*exp(x)*x+x)+exp(3),x,method=_RETURNVERBOSE)

[Out]

exp(4*exp(x)*x+x)+x*exp(3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \left (e^3+e^{x+4 e^x x} \left (1+e^x (4+4 x)\right )\right ) \, dx=x e^{3} + e^{\left (4 \, x e^{x} + x\right )} \]

[In]

integrate(((4+4*x)*exp(x)+1)*exp(4*exp(x)*x+x)+exp(3),x, algorithm="fricas")

[Out]

x*e^3 + e^(4*x*e^x + x)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \left (e^3+e^{x+4 e^x x} \left (1+e^x (4+4 x)\right )\right ) \, dx=x e^{3} + e^{4 x e^{x} + x} \]

[In]

integrate(((4+4*x)*exp(x)+1)*exp(4*exp(x)*x+x)+exp(3),x)

[Out]

x*exp(3) + exp(4*x*exp(x) + x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \left (e^3+e^{x+4 e^x x} \left (1+e^x (4+4 x)\right )\right ) \, dx=x e^{3} + e^{\left (4 \, x e^{x} + x\right )} \]

[In]

integrate(((4+4*x)*exp(x)+1)*exp(4*exp(x)*x+x)+exp(3),x, algorithm="maxima")

[Out]

x*e^3 + e^(4*x*e^x + x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \left (e^3+e^{x+4 e^x x} \left (1+e^x (4+4 x)\right )\right ) \, dx=x e^{3} + e^{\left (4 \, x e^{x} + x\right )} \]

[In]

integrate(((4+4*x)*exp(x)+1)*exp(4*exp(x)*x+x)+exp(3),x, algorithm="giac")

[Out]

x*e^3 + e^(4*x*e^x + x)

Mupad [B] (verification not implemented)

Time = 11.15 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \left (e^3+e^{x+4 e^x x} \left (1+e^x (4+4 x)\right )\right ) \, dx={\mathrm {e}}^{x+4\,x\,{\mathrm {e}}^x}+x\,{\mathrm {e}}^3 \]

[In]

int(exp(3) + exp(x + 4*x*exp(x))*(exp(x)*(4*x + 4) + 1),x)

[Out]

exp(x + 4*x*exp(x)) + x*exp(3)