3.103.1 \(\int e^x (6+6 x) \, dx\)

Optimal. Leaf size=15 \[ -2-e^{e^8}+6 e^x x \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.93, 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} \begin {gather*} 6 e^x (x+1)-6 e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*(6 + 6*x),x]

[Out]

-6*E^x + 6*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} &=6 e^x (1+x)-6 \int e^x \, dx\\ &=-6 e^x+6 e^x (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 6, normalized size = 0.40 \begin {gather*} 6 e^x x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*(6 + 6*x),x]

[Out]

6*E^x*x

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fricas [A]  time = 0.68, size = 5, normalized size = 0.33 \begin {gather*} 6 \, x e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x+6)*exp(x),x, algorithm="fricas")

[Out]

6*x*e^x

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giac [A]  time = 0.21, size = 5, normalized size = 0.33 \begin {gather*} 6 \, x e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x+6)*exp(x),x, algorithm="giac")

[Out]

6*x*e^x

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maple [A]  time = 0.05, size = 6, normalized size = 0.40

method	result	size
gosper	\(6 \,{\mathrm e}^{x} x\)	\(6\)
default	\(6 \,{\mathrm e}^{x} x\)	\(6\)
norman	\(6 \,{\mathrm e}^{x} x\)	\(6\)
risch	\(6 \,{\mathrm e}^{x} x\)	\(6\)
meijerg	\(-3 \left (-2 x +2\right ) {\mathrm e}^{x}+6 \,{\mathrm e}^{x}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x+6)*exp(x),x,method=_RETURNVERBOSE)

[Out]

6*exp(x)*x

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maxima [A]  time = 0.36, size = 12, normalized size = 0.80 \begin {gather*} 6 \, {\left (x - 1\right )} e^{x} + 6 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x+6)*exp(x),x, algorithm="maxima")

[Out]

6*(x - 1)*e^x + 6*e^x

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mupad [B]  time = 0.04, size = 5, normalized size = 0.33 \begin {gather*} 6\,x\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(6*x + 6),x)

[Out]

6*x*exp(x)

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sympy [A]  time = 0.08, size = 5, normalized size = 0.33 \begin {gather*} 6 x e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x+6)*exp(x),x)

[Out]

6*x*exp(x)

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