3.80.96 \(\int e^x (1+x) \, dx\)

Optimal. Leaf size=7 \[ -9+e^x x \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.86, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2176, 2194} \begin {gather*} e^x (x+1)-e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*(1 + x),x]

[Out]

-E^x + E^x*(1 + 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} &=e^x (1+x)-\int e^x \, dx\\ &=-e^x+e^x (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 5, normalized size = 0.71 \begin {gather*} e^x x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*(1 + x),x]

[Out]

E^x*x

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fricas [A]  time = 0.78, size = 4, normalized size = 0.57 \begin {gather*} x e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^x

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giac [A]  time = 0.12, size = 4, normalized size = 0.57 \begin {gather*} x e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^x

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maple [A]  time = 0.02, size = 5, normalized size = 0.71

method	result	size
gosper	\({\mathrm e}^{x} x\)	\(5\)
default	\({\mathrm e}^{x} x\)	\(5\)
norman	\({\mathrm e}^{x} x\)	\(5\)
risch	\({\mathrm e}^{x} x\)	\(5\)
meijerg	\(-\frac {\left (-2 x +2\right ) {\mathrm e}^{x}}{2}+{\mathrm e}^{x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)*x

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maxima [A]  time = 0.37, size = 9, normalized size = 1.29 \begin {gather*} {\left (x - 1\right )} e^{x} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x - 1)*e^x + e^x

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mupad [B]  time = 0.02, size = 4, normalized size = 0.57 \begin {gather*} x\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(x + 1),x)

[Out]

x*exp(x)

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sympy [A]  time = 0.07, size = 3, normalized size = 0.43 \begin {gather*} x e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(x)

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