3.1.46 \(\int e^x (3+2 x) \, dx\) [46]

Optimal. Leaf size=15 \[ -2 e^x+e^x (3+2 x) \]

[Out]

-2*exp(x)+exp(x)*(3+2*x)

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2207, 2225} \begin {gather*} e^x (2 x+3)-2 e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*(3 + 2*x),x]

[Out]

-2*E^x + E^x*(3 + 2*x)

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 {align*} \int e^x (3+2 x) \, dx &=e^x (3+2 x)-2 \int e^x \, dx\\ &=-2 e^x+e^x (3+2 x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 9, normalized size = 0.60 \begin {gather*} e^x (1+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*(3 + 2*x),x]

[Out]

E^x*(1 + 2*x)

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Mathics [A]
time = 1.72, size = 9, normalized size = 0.60 \begin {gather*} \left (1+2 x\right ) E^x \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(3 + 2*x)*E^x,x]')

[Out]

(1 + 2 x) E ^ x

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Maple [A]
time = 0.01, size = 9, normalized size = 0.60

method	result	size
gosper	\(\left (1+2 x \right ) {\mathrm e}^{x}\)	\(9\)
default	\(2 \,{\mathrm e}^{x} x +{\mathrm e}^{x}\)	\(9\)
norman	\(2 \,{\mathrm e}^{x} x +{\mathrm e}^{x}\)	\(9\)
risch	\(\left (1+2 x \right ) {\mathrm e}^{x}\)	\(9\)
meijerg	\(-1+3 \,{\mathrm e}^{x}-\left (-2 x +2\right ) {\mathrm e}^{x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(3+2*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 12, normalized size = 0.80 \begin {gather*} 2 \, {\left (x - 1\right )} e^{x} + 3 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(3+2*x),x, algorithm="maxima")

[Out]

2*(x - 1)*e^x + 3*e^x

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Fricas [A]
time = 0.32, size = 8, normalized size = 0.53 \begin {gather*} {\left (2 \, x + 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(3+2*x),x, algorithm="fricas")

[Out]

(2*x + 1)*e^x

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Sympy [A]
time = 0.04, size = 7, normalized size = 0.47 \begin {gather*} \left (2 x + 1\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(3+2*x),x)

[Out]

(2*x + 1)*exp(x)

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Giac [A]
time = 0.00, size = 8, normalized size = 0.53 \begin {gather*} \left (2 x+1\right ) \mathrm {e}^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(3+2*x),x)

[Out]

e^x*(2*x + 1)

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Mupad [B]
time = 0.03, size = 8, normalized size = 0.53 \begin {gather*} {\mathrm {e}}^x\,\left (2\,x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(2*x + 3),x)

[Out]

exp(x)*(2*x + 1)

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