3.71.32 \(\int e^{4 x+x^2} (4+2 x) \, dx\)

Optimal. Leaf size=9 \[ e^{4 x+x^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2236} \begin {gather*} e^{x^2+4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4*x + x^2)*(4 + 2*x),x]

[Out]

E^(4*x + x^2)

Rule 2236

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 {gather*} \begin {aligned} \text {integral} &=e^{4 x+x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 9, normalized size = 1.00 \begin {gather*} e^{4 x+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4*x + x^2)*(4 + 2*x),x]

[Out]

E^(4*x + x^2)

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fricas [A]  time = 0.53, size = 8, normalized size = 0.89 \begin {gather*} e^{\left (x^{2} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^2 + 4*x)

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giac [A]  time = 0.13, size = 8, normalized size = 0.89 \begin {gather*} e^{\left (x^{2} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^2 + 4*x)

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maple [A]  time = 0.03, size = 7, normalized size = 0.78

method	result	size
risch	\({\mathrm e}^{\left (4+x \right ) x}\)	\(7\)
gosper	\({\mathrm e}^{x^{2}+4 x}\)	\(9\)
derivativedivides	\({\mathrm e}^{x^{2}+4 x}\)	\(9\)
default	\({\mathrm e}^{x^{2}+4 x}\)	\(9\)
norman	\({\mathrm e}^{x^{2}+4 x}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp((4+x)*x)

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maxima [A]  time = 0.38, size = 8, normalized size = 0.89 \begin {gather*} e^{\left (x^{2} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^2 + 4*x)

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mupad [B]  time = 4.13, size = 6, normalized size = 0.67 \begin {gather*} {\mathrm {e}}^{x\,\left (x+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4*x + x^2)*(2*x + 4),x)

[Out]

exp(x*(x + 4))

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sympy [A]  time = 0.08, size = 7, normalized size = 0.78 \begin {gather*} e^{x^{2} + 4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+4)*exp(x**2+4*x),x)

[Out]

exp(x**2 + 4*x)

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