3.65.62 \(\int (e^3+e^{x+4 e^x x} (1+e^x (4+4 x))) \, dx\)

Optimal. Leaf size=18 \[ e^{x+4 e^x x}+e^3 (5+x) \]

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Rubi [A]  time = 0.06, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6706} \begin {gather*} e^3 x+e^{4 e^x x+x} \end {gather*}

Antiderivative was successfully verified.

[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 6706

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 16, normalized size = 0.89 \begin {gather*} e^{x+4 e^x x}+e^3 x \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.59, size = 13, normalized size = 0.72 \begin {gather*} x e^{3} + e^{\left (4 \, x e^{x} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^3 + e^(4*x*e^x + x)

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giac [A]  time = 0.12, size = 13, normalized size = 0.72 \begin {gather*} x e^{3} + e^{\left (4 \, x e^{x} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^3 + e^(4*x*e^x + x)

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maple [A]  time = 0.07, size = 14, normalized size = 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	\({\mathrm e}^{x \left (4 \,{\mathrm e}^{x}+1\right )}+x \,{\mathrm e}^{3}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 13, normalized size = 0.72 \begin {gather*} x e^{3} + e^{\left (4 \, x e^{x} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^3 + e^(4*x*e^x + x)

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mupad [B]  time = 0.07, size = 13, normalized size = 0.72 \begin {gather*} {\mathrm {e}}^{x+4\,x\,{\mathrm {e}}^x}+x\,{\mathrm {e}}^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.15, size = 14, normalized size = 0.78 \begin {gather*} x e^{3} + e^{4 x e^{x} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(3) + exp(4*x*exp(x) + x)

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