3.45.73 \(\int (1+e^x (2+x)) \, dx\)

Optimal. Leaf size=11 \[ 4+e^x+x+e^x x \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.27, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2176, 2194} \begin {gather*} x-e^x+e^x (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^x*(2 + x),x]

[Out]

-E^x + x + E^x*(2 + x)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+\int e^x (2+x) \, dx\\ &=x+e^x (2+x)-\int e^x \, dx\\ &=-e^x+x+e^x (2+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 0.82 \begin {gather*} x+e^x (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^x*(2 + x),x]

[Out]

x + E^x*(1 + x)

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fricas [A]  time = 0.58, size = 8, normalized size = 0.73 \begin {gather*} {\left (x + 1\right )} e^{x} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*exp(x)+1,x, algorithm="fricas")

[Out]

(x + 1)*e^x + x

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giac [A]  time = 0.26, size = 8, normalized size = 0.73 \begin {gather*} {\left (x + 1\right )} e^{x} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*exp(x)+1,x, algorithm="giac")

[Out]

(x + 1)*e^x + x

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maple [A]  time = 0.01, size = 9, normalized size = 0.82

method	result	size
default	\(x +{\mathrm e}^{x} x +{\mathrm e}^{x}\)	\(9\)
norman	\(x +{\mathrm e}^{x} x +{\mathrm e}^{x}\)	\(9\)
risch	\(\left (x +1\right ) {\mathrm e}^{x}+x\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+x)*exp(x)+1,x,method=_RETURNVERBOSE)

[Out]

x+exp(x)*x+exp(x)

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maxima [A]  time = 0.35, size = 12, normalized size = 1.09 \begin {gather*} {\left (x - 1\right )} e^{x} + x + 2 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*exp(x)+1,x, algorithm="maxima")

[Out]

(x - 1)*e^x + x + 2*e^x

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mupad [B]  time = 3.08, size = 8, normalized size = 0.73 \begin {gather*} x+{\mathrm {e}}^x+x\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(x + 2) + 1,x)

[Out]

x + exp(x) + x*exp(x)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.64 \begin {gather*} x + \left (x + 1\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*exp(x)+1,x)

[Out]

x + (x + 1)*exp(x)

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