3.102.53 \(\int (4-2 x+3 x^2+e (-2+4 x)) \, dx\)

Optimal. Leaf size=25 \[ e^4-e^{e^2}+(2 e+x) \left (4-x+x^2\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 0, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} x^3-x^2+4 x+\frac {1}{2} e (1-2 x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4 - 2*x + 3*x^2 + E*(-2 + 4*x),x]

[Out]

(E*(1 - 2*x)^2)/2 + 4*x - x^2 + x^3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} e (1-2 x)^2+4 x-x^2+x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 0.88 \begin {gather*} 4 x-2 e x-x^2+2 e x^2+x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4 - 2*x + 3*x^2 + E*(-2 + 4*x),x]

[Out]

4*x - 2*E*x - x^2 + 2*E*x^2 + x^3

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fricas [A]  time = 0.59, size = 23, normalized size = 0.92 \begin {gather*} x^{3} - x^{2} + 2 \, {\left (x^{2} - x\right )} e + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-2)*exp(1)+3*x^2-2*x+4,x, algorithm="fricas")

[Out]

x^3 - x^2 + 2*(x^2 - x)*e + 4*x

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giac [A]  time = 0.13, size = 23, normalized size = 0.92 \begin {gather*} x^{3} - x^{2} + 2 \, {\left (x^{2} - x\right )} e + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-2)*exp(1)+3*x^2-2*x+4,x, algorithm="giac")

[Out]

x^3 - x^2 + 2*(x^2 - x)*e + 4*x

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maple [A]  time = 0.02, size = 20, normalized size = 0.80

method	result	size
gosper	\(x \left (2 x \,{\mathrm e}+x^{2}-2 \,{\mathrm e}-x +4\right )\)	\(20\)
norman	\(x^{3}+\left (2 \,{\mathrm e}-1\right ) x^{2}+\left (-2 \,{\mathrm e}+4\right ) x\)	\(23\)
default	\({\mathrm e} \left (2 x^{2}-2 x \right )+x^{3}-x^{2}+4 x\)	\(25\)
risch	\(2 x^{2} {\mathrm e}-2 x \,{\mathrm e}+x^{3}-x^{2}+4 x\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x-2)*exp(1)+3*x^2-2*x+4,x,method=_RETURNVERBOSE)

[Out]

x*(2*x*exp(1)+x^2-2*exp(1)-x+4)

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maxima [A]  time = 0.35, size = 23, normalized size = 0.92 \begin {gather*} x^{3} - x^{2} + 2 \, {\left (x^{2} - x\right )} e + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-2)*exp(1)+3*x^2-2*x+4,x, algorithm="maxima")

[Out]

x^3 - x^2 + 2*(x^2 - x)*e + 4*x

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mupad [B]  time = 7.22, size = 23, normalized size = 0.92 \begin {gather*} x^3+\left (2\,\mathrm {e}-1\right )\,x^2+\left (4-2\,\mathrm {e}\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*x^2 - 2*x + exp(1)*(4*x - 2) + 4,x)

[Out]

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

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sympy [A]  time = 0.05, size = 20, normalized size = 0.80 \begin {gather*} x^{3} + x^{2} \left (-1 + 2 e\right ) + x \left (4 - 2 e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-2)*exp(1)+3*x**2-2*x+4,x)

[Out]

x**3 + x**2*(-1 + 2*E) + x*(4 - 2*E)

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