3.10.7 \(\int (-12 e x^2+e^{1+x} (4+4 x)) \, dx\)

Optimal. Leaf size=26 \[ e \left (-4+4 \left (1+\frac {\frac {e^x}{x}-2 x}{x}\right ) x^3\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2176, 2194} \begin {gather*} -4 e x^3-4 e^{x+1}+4 e^{x+1} (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-12*E*x^2 + E^(1 + x)*(4 + 4*x),x]

[Out]

-4*E^(1 + x) - 4*E*x^3 + 4*E^(1 + 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} &=-4 e x^3+\int e^{1+x} (4+4 x) \, dx\\ &=-4 e x^3+4 e^{1+x} (1+x)-4 \int e^{1+x} \, dx\\ &=-4 e^{1+x}-4 e x^3+4 e^{1+x} (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.54 \begin {gather*} 4 e \left (e^x x-x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-12*E*x^2 + E^(1 + x)*(4 + 4*x),x]

[Out]

4*E*(E^x*x - x^3)

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fricas [A]  time = 0.72, size = 15, normalized size = 0.58 \begin {gather*} -4 \, x^{3} e + 4 \, x e^{\left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+4)*exp(1)*exp(x)-12*x^2*exp(1),x, algorithm="fricas")

[Out]

-4*x^3*e + 4*x*e^(x + 1)

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giac [A]  time = 0.38, size = 15, normalized size = 0.58 \begin {gather*} -4 \, x^{3} e + 4 \, x e^{\left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+4)*exp(1)*exp(x)-12*x^2*exp(1),x, algorithm="giac")

[Out]

-4*x^3*e + 4*x*e^(x + 1)

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maple [A]  time = 0.04, size = 16, normalized size = 0.62

method	result	size
default	\(4 x \,{\mathrm e} \,{\mathrm e}^{x}-4 x^{3} {\mathrm e}\)	\(16\)
norman	\(4 x \,{\mathrm e} \,{\mathrm e}^{x}-4 x^{3} {\mathrm e}\)	\(16\)
risch	\(4 x \,{\mathrm e}^{x +1}-4 x^{3} {\mathrm e}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x+4)*exp(1)*exp(x)-12*x^2*exp(1),x,method=_RETURNVERBOSE)

[Out]

4*x*exp(1)*exp(x)-4*x^3*exp(1)

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maxima [A]  time = 0.36, size = 27, normalized size = 1.04 \begin {gather*} -4 \, x^{3} e + 4 \, {\left (x e - e\right )} e^{x} + 4 \, e^{\left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+4)*exp(1)*exp(x)-12*x^2*exp(1),x, algorithm="maxima")

[Out]

-4*x^3*e + 4*(x*e - e)*e^x + 4*e^(x + 1)

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mupad [B]  time = 0.05, size = 13, normalized size = 0.50 \begin {gather*} 4\,x\,\mathrm {e}\,\left ({\mathrm {e}}^x-x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1)*exp(x)*(4*x + 4) - 12*x^2*exp(1),x)

[Out]

4*x*exp(1)*(exp(x) - x^2)

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sympy [A]  time = 0.09, size = 17, normalized size = 0.65 \begin {gather*} - 4 e x^{3} + 4 e x e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+4)*exp(1)*exp(x)-12*x**2*exp(1),x)

[Out]

-4*E*x**3 + 4*E*x*exp(x)

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