3.914 \(\int \frac{\sqrt{2+e x}}{\sqrt{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=20 \[ -\frac{2 \sqrt{2-e x}}{\sqrt{3} e} \]

[Out]

(-2*Sqrt[2 - e*x])/(Sqrt[3]*e)

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Rubi [A]  time = 0.0085675, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {627, 32} \[ -\frac{2 \sqrt{2-e x}}{\sqrt{3} e} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + e*x]/Sqrt[12 - 3*e^2*x^2],x]

[Out]

(-2*Sqrt[2 - e*x])/(Sqrt[3]*e)

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{2+e x}}{\sqrt{12-3 e^2 x^2}} \, dx &=\int \frac{1}{\sqrt{6-3 e x}} \, dx\\ &=-\frac{2 \sqrt{2-e x}}{\sqrt{3} e}\\ \end{align*}

Mathematica [A]  time = 0.0347937, size = 33, normalized size = 1.65 \[ \frac{2 (e x-2) \sqrt{e x+2}}{e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + e*x]/Sqrt[12 - 3*e^2*x^2],x]

[Out]

(2*(-2 + e*x)*Sqrt[2 + e*x])/(e*Sqrt[12 - 3*e^2*x^2])

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Maple [A]  time = 0.039, size = 30, normalized size = 1.5 \begin{align*} 2\,{\frac{ \left ( ex-2 \right ) \sqrt{ex+2}}{e\sqrt{-3\,{e}^{2}{x}^{2}+12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/2),x)

[Out]

2*(e*x-2)*(e*x+2)^(1/2)/e/(-3*e^2*x^2+12)^(1/2)

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Maxima [C]  time = 1.73152, size = 34, normalized size = 1.7 \begin{align*} -\frac{2 i \, \sqrt{3} e x - 4 i \, \sqrt{3}}{3 \, \sqrt{e x - 2} e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="maxima")

[Out]

-1/3*(2*I*sqrt(3)*e*x - 4*I*sqrt(3))/(sqrt(e*x - 2)*e)

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Fricas [A]  time = 1.75109, size = 76, normalized size = 3.8 \begin{align*} -\frac{2 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{3 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^2*x + 2*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{3} \int \frac{\sqrt{e x + 2}}{\sqrt{- e^{2} x^{2} + 4}}\, dx}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)**(1/2)/(-3*e**2*x**2+12)**(1/2),x)

[Out]

sqrt(3)*Integral(sqrt(e*x + 2)/sqrt(-e**2*x**2 + 4), x)/3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + 2}}{\sqrt{-3 \, e^{2} x^{2} + 12}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(e*x + 2)/sqrt(-3*e^2*x^2 + 12), x)