3.53.90 \(\int (8-4 e^{2 e}-14 x-32 e^e x-48 x^2) \, dx\)

Optimal. Leaf size=22 \[ x \left (x+4 \left (2-2 x-\left (e^e+2 x\right )^2\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6} \begin {gather*} -16 x^3-\left (7+16 e^e\right ) x^2+4 \left (2-e^{2 e}\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[8 - 4*E^(2*E) - 14*x - 32*E^E*x - 48*x^2,x]

[Out]

4*(2 - E^(2*E))*x - (7 + 16*E^E)*x^2 - 16*x^3

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (8-4 e^{2 e}+\left (-14-32 e^e\right ) x-48 x^2\right ) \, dx\\ &=4 \left (2-e^{2 e}\right ) x-\left (7+16 e^e\right ) x^2-16 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.36 \begin {gather*} 8 x-4 e^{2 e} x-7 x^2-16 e^e x^2-16 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[8 - 4*E^(2*E) - 14*x - 32*E^E*x - 48*x^2,x]

[Out]

8*x - 4*E^(2*E)*x - 7*x^2 - 16*E^E*x^2 - 16*x^3

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fricas [A]  time = 0.50, size = 30, normalized size = 1.36 \begin {gather*} -16 \, x^{3} - 16 \, x^{2} e^{e} - 7 \, x^{2} - 4 \, x e^{\left (2 \, e\right )} + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(exp(1))^2-32*x*exp(exp(1))-48*x^2-14*x+8,x, algorithm="fricas")

[Out]

-16*x^3 - 16*x^2*e^e - 7*x^2 - 4*x*e^(2*e) + 8*x

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giac [A]  time = 0.19, size = 30, normalized size = 1.36 \begin {gather*} -16 \, x^{3} - 16 \, x^{2} e^{e} - 7 \, x^{2} - 4 \, x e^{\left (2 \, e\right )} + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(exp(1))^2-32*x*exp(exp(1))-48*x^2-14*x+8,x, algorithm="giac")

[Out]

-16*x^3 - 16*x^2*e^e - 7*x^2 - 4*x*e^(2*e) + 8*x

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maple [A]  time = 0.03, size = 27, normalized size = 1.23

method	result	size
gosper	\(-x \left (4 \,{\mathrm e}^{2 \,{\mathrm e}}+16 x \,{\mathrm e}^{{\mathrm e}}+16 x^{2}+7 x -8\right )\)	\(27\)
norman	\(\left (-4 \,{\mathrm e}^{2 \,{\mathrm e}}+8\right ) x +\left (-16 \,{\mathrm e}^{{\mathrm e}}-7\right ) x^{2}-16 x^{3}\)	\(29\)
default	\(-4 \,{\mathrm e}^{2 \,{\mathrm e}} x -16 x^{2} {\mathrm e}^{{\mathrm e}}-16 x^{3}-7 x^{2}+8 x\)	\(31\)
risch	\(-4 \,{\mathrm e}^{2 \,{\mathrm e}} x -16 x^{2} {\mathrm e}^{{\mathrm e}}-16 x^{3}-7 x^{2}+8 x\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-4*exp(exp(1))^2-32*x*exp(exp(1))-48*x^2-14*x+8,x,method=_RETURNVERBOSE)

[Out]

-x*(4*exp(exp(1))^2+16*x*exp(exp(1))+16*x^2+7*x-8)

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maxima [A]  time = 0.37, size = 30, normalized size = 1.36 \begin {gather*} -16 \, x^{3} - 16 \, x^{2} e^{e} - 7 \, x^{2} - 4 \, x e^{\left (2 \, e\right )} + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(exp(1))^2-32*x*exp(exp(1))-48*x^2-14*x+8,x, algorithm="maxima")

[Out]

-16*x^3 - 16*x^2*e^e - 7*x^2 - 4*x*e^(2*e) + 8*x

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mupad [B]  time = 3.41, size = 30, normalized size = 1.36 \begin {gather*} -16\,x^3+\left (-16\,{\mathrm {e}}^{\mathrm {e}}-7\right )\,x^2+\left (8-4\,{\mathrm {e}}^{2\,\mathrm {e}}\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8 - 4*exp(2*exp(1)) - 32*x*exp(exp(1)) - 48*x^2 - 14*x,x)

[Out]

- x*(4*exp(2*exp(1)) - 8) - x^2*(16*exp(exp(1)) + 7) - 16*x^3

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sympy [A]  time = 0.06, size = 29, normalized size = 1.32 \begin {gather*} - 16 x^{3} + x^{2} \left (- 16 e^{e} - 7\right ) + x \left (8 - 4 e^{2 e}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(exp(1))**2-32*x*exp(exp(1))-48*x**2-14*x+8,x)

[Out]

-16*x**3 + x**2*(-16*exp(E) - 7) + x*(8 - 4*exp(2*E))

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