3.3.76 \(\int (-1+e^x (9+3 x)) \, dx\)

Optimal. Leaf size=12 \[ -x+3 e^x (2+x) \]

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.42, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2176, 2194} \begin {gather*} -x-3 e^x+3 e^x (x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + E^x*(9 + 3*x),x]

[Out]

-3*E^x - x + 3*E^x*(3 + 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 (9+3 x) \, dx\\ &=-x+3 e^x (3+x)-3 \int e^x \, dx\\ &=-3 e^x-x+3 e^x (3+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} -x+3 e^x (2+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + E^x*(9 + 3*x),x]

[Out]

-x + 3*E^x*(2 + x)

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fricas [A]  time = 1.10, size = 11, normalized size = 0.92 \begin {gather*} 3 \, {\left (x + 2\right )} e^{x} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+9)*exp(x)-1,x, algorithm="fricas")

[Out]

3*(x + 2)*e^x - x

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giac [A]  time = 0.78, size = 11, normalized size = 0.92 \begin {gather*} 3 \, {\left (x + 2\right )} e^{x} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+9)*exp(x)-1,x, algorithm="giac")

[Out]

3*(x + 2)*e^x - x

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maple [A]  time = 0.02, size = 13, normalized size = 1.08

method	result	size
risch	\({\mathrm e}^{x} \left (6+3 x \right )-x\)	\(13\)
default	\(-x +3 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x}\)	\(14\)
norman	\(-x +3 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+9)*exp(x)-1,x,method=_RETURNVERBOSE)

[Out]

exp(x)*(6+3*x)-x

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maxima [A]  time = 0.51, size = 15, normalized size = 1.25 \begin {gather*} 3 \, {\left (x - 1\right )} e^{x} - x + 9 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+9)*exp(x)-1,x, algorithm="maxima")

[Out]

3*(x - 1)*e^x - x + 9*e^x

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mupad [B]  time = 0.30, size = 13, normalized size = 1.08 \begin {gather*} 6\,{\mathrm {e}}^x-x+3\,x\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(3*x + 9) - 1,x)

[Out]

6*exp(x) - x + 3*x*exp(x)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.67 \begin {gather*} - x + \left (3 x + 6\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+9)*exp(x)-1,x)

[Out]

-x + (3*x + 6)*exp(x)

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