3.101.44 \(\int (2-4 x+4 e^2 x+6 x^2) \, dx\)

Optimal. Leaf size=19 \[ 2 x \left (1+x+e^2 x-(2-x) x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6} \begin {gather*} 2 x^3-2 \left (1-e^2\right ) x^2+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 - 4*x + 4*E^2*x + 6*x^2,x]

[Out]

2*x - 2*(1 - E^2)*x^2 + 2*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 (2+\left (-4+4 e^2\right ) x+6 x^2\right ) \, dx\\ &=2 x-2 \left (1-e^2\right ) x^2+2 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.16 \begin {gather*} 2 x-2 x^2+2 e^2 x^2+2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 - 4*x + 4*E^2*x + 6*x^2,x]

[Out]

2*x - 2*x^2 + 2*E^2*x^2 + 2*x^3

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fricas [A]  time = 0.64, size = 21, normalized size = 1.11 \begin {gather*} 2 \, x^{3} + 2 \, x^{2} e^{2} - 2 \, x^{2} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(2)*x+6*x^2-4*x+2,x, algorithm="fricas")

[Out]

2*x^3 + 2*x^2*e^2 - 2*x^2 + 2*x

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giac [A]  time = 0.19, size = 21, normalized size = 1.11 \begin {gather*} 2 \, x^{3} + 2 \, x^{2} e^{2} - 2 \, x^{2} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(2)*x+6*x^2-4*x+2,x, algorithm="giac")

[Out]

2*x^3 + 2*x^2*e^2 - 2*x^2 + 2*x

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maple [A]  time = 0.02, size = 16, normalized size = 0.84

method	result	size
gosper	\(2 x \left ({\mathrm e}^{2} x +x^{2}-x +1\right )\)	\(16\)
norman	\(\left (2 \,{\mathrm e}^{2}-2\right ) x^{2}+2 x +2 x^{3}\)	\(20\)
default	\(2 x^{2} {\mathrm e}^{2}+2 x^{3}-2 x^{2}+2 x\)	\(22\)
risch	\(2 x^{2} {\mathrm e}^{2}+2 x^{3}-2 x^{2}+2 x\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*exp(2)*x+6*x^2-4*x+2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 21, normalized size = 1.11 \begin {gather*} 2 \, x^{3} + 2 \, x^{2} e^{2} - 2 \, x^{2} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(2)*x+6*x^2-4*x+2,x, algorithm="maxima")

[Out]

2*x^3 + 2*x^2*e^2 - 2*x^2 + 2*x

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mupad [B]  time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} 2\,x^3+\left (2\,{\mathrm {e}}^2-2\right )\,x^2+2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x*exp(2) - 4*x + 6*x^2 + 2,x)

[Out]

2*x + x^2*(2*exp(2) - 2) + 2*x^3

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sympy [A]  time = 0.05, size = 17, normalized size = 0.89 \begin {gather*} 2 x^{3} + x^{2} \left (-2 + 2 e^{2}\right ) + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(2)*x+6*x**2-4*x+2,x)

[Out]

2*x**3 + x**2*(-2 + 2*exp(2)) + 2*x

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