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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, 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 = {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 = 0.01, size = 9, normalized size = 0.60 \[ e^x (2 x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*(3 + 2*x),x]

[Out]

E^x*(1 + 2*x)

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 8, normalized size = 0.53 \[ {\left (2 \, x + 1\right )} e^{x} \]

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

________________________________________________________________________________________

giac [A]  time = 0.72, size = 8, normalized size = 0.53 \[ {\left (2 \, x + 1\right )} e^{x} \]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 9, normalized size = 0.60 \[ \left (2 x +1\right ) {\mathrm e}^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 12, normalized size = 0.80 \[ 2 \, {\left (x - 1\right )} e^{x} + 3 \, e^{x} \]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(2*x + 3),x)

[Out]

exp(x)*(2*x + 1)

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 7, normalized size = 0.47 \[ \left (2 x + 1\right ) e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x + 1)*exp(x)

________________________________________________________________________________________