3.46 \(\int e^x (3+2 x) \, dx\)

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

[Out]

-2*E^x + E^x*(3 + 2*x)

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Rubi [A]  time = 0.0083929, antiderivative size = 15, normalized size of antiderivative = 1., 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 = {2176, 2194} \[ e^x (2 x+3)-2 e^x \]

Antiderivative was successfully verified.

[In]

Int[E^x*(3 + 2*x),x]

[Out]

-2*E^x + E^x*(3 + 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{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*}

Mathematica [A]  time = 0.009235, size = 9, normalized size = 0.6 \[ e^x (2 x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*(3 + 2*x),x]

[Out]

E^x*(1 + 2*x)

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Maple [A]  time = 0., size = 9, normalized size = 0.6 \begin{align*} \left ( 1+2\,x \right ){{\rm e}^{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.944416, size = 16, normalized size = 1.07 \begin{align*} 2 \,{\left (x - 1\right )} e^{x} + 3 \, e^{x} \end{align*}

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 = 2.2422, size = 20, normalized size = 1.33 \begin{align*}{\left (2 \, x + 1\right )} e^{x} \end{align*}

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 = 0.074453, size = 7, normalized size = 0.47 \begin{align*} \left (2 x + 1\right ) e^{x} \end{align*}

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 = 1.05474, size = 11, normalized size = 0.73 \begin{align*}{\left (2 \, x + 1\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x + 1)*e^x