3.21.13 \(\int (1+4 x+\sqrt {e} (-6 x-4 x^3)) \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 0, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} -\sqrt {e} x^4-3 \sqrt {e} x^2+2 x^2+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + 4*x + Sqrt[E]*(-6*x - 4*x^3),x]

[Out]

x + 2*x^2 - 3*Sqrt[E]*x^2 - Sqrt[E]*x^4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+2 x^2+\sqrt {e} \int \left (-6 x-4 x^3\right ) \, dx\\ &=x+2 x^2-3 \sqrt {e} x^2-\sqrt {e} x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.04 \begin {gather*} x+2 x^2-3 \sqrt {e} x^2-\sqrt {e} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + 4*x + Sqrt[E]*(-6*x - 4*x^3),x]

[Out]

x + 2*x^2 - 3*Sqrt[E]*x^2 - Sqrt[E]*x^4

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fricas [A]  time = 0.48, size = 20, normalized size = 0.77 \begin {gather*} 2 \, x^{2} - {\left (x^{4} + 3 \, x^{2}\right )} e^{\frac {1}{2}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^3-6*x)*exp(1/4)^2+4*x+1,x, algorithm="fricas")

[Out]

2*x^2 - (x^4 + 3*x^2)*e^(1/2) + x

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giac [A]  time = 0.26, size = 20, normalized size = 0.77 \begin {gather*} 2 \, x^{2} - {\left (x^{4} + 3 \, x^{2}\right )} e^{\frac {1}{2}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^3-6*x)*exp(1/4)^2+4*x+1,x, algorithm="giac")

[Out]

2*x^2 - (x^4 + 3*x^2)*e^(1/2) + x

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maple [A]  time = 0.03, size = 22, normalized size = 0.85

method	result	size
risch	\(-{\mathrm e}^{\frac {1}{2}} x^{4}-3 x^{2} {\mathrm e}^{\frac {1}{2}}+2 x^{2}+x\)	\(22\)
gosper	\(-x \left ({\mathrm e}^{\frac {1}{2}} x^{3}+3 x \,{\mathrm e}^{\frac {1}{2}}-2 x -1\right )\)	\(24\)
default	\({\mathrm e}^{\frac {1}{2}} \left (-x^{4}-3 x^{2}\right )+2 x^{2}+x\)	\(24\)
norman	\(x +\left (-3 \,{\mathrm e}^{\frac {1}{2}}+2\right ) x^{2}-{\mathrm e}^{\frac {1}{2}} x^{4}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^3-6*x)*exp(1/4)^2+4*x+1,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.34, size = 20, normalized size = 0.77 \begin {gather*} 2 \, x^{2} - {\left (x^{4} + 3 \, x^{2}\right )} e^{\frac {1}{2}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^3-6*x)*exp(1/4)^2+4*x+1,x, algorithm="maxima")

[Out]

2*x^2 - (x^4 + 3*x^2)*e^(1/2) + x

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mupad [B]  time = 1.10, size = 20, normalized size = 0.77 \begin {gather*} -\sqrt {\mathrm {e}}\,x^4+\left (2-3\,\sqrt {\mathrm {e}}\right )\,x^2+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x - exp(1/2)*(6*x + 4*x^3) + 1,x)

[Out]

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

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sympy [A]  time = 0.06, size = 20, normalized size = 0.77 \begin {gather*} - x^{4} e^{\frac {1}{2}} + x^{2} \left (2 - 3 e^{\frac {1}{2}}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**3-6*x)*exp(1/4)**2+4*x+1,x)

[Out]

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

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