3.296 \(\int \frac{\sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]

[Out]

-(EllipticE[ArcCos[Sqrt[3/2]*x], 2]/Sqrt[3])

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Rubi [A]  time = 0.0065772, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {425} \[ -\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + 3*x^2]/Sqrt[2 - 3*x^2],x]

[Out]

-(EllipticE[ArcCos[Sqrt[3/2]*x], 2]/Sqrt[3])

Rule 425

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> -Simp[(Sqrt[a - (b*c)/d]*EllipticE[ArcCo
s[Rt[-(d/c), 2]*x], (b*c)/(b*c - a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&
GtQ[c, 0] && GtQ[a - (b*c)/d, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx &=-\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0259826, size = 35, normalized size = 1.84 \[ \frac{\sqrt{3 x^2-1} E\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3-9 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + 3*x^2]/Sqrt[2 - 3*x^2],x]

[Out]

(Sqrt[-1 + 3*x^2]*EllipticE[ArcSin[Sqrt[3/2]*x], 2])/Sqrt[3 - 9*x^2]

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Maple [A]  time = 0.014, size = 37, normalized size = 2. \begin{align*} -{\frac{\sqrt{3}}{3}{\it EllipticE} \left ({\frac{x\sqrt{2}\sqrt{3}}{2}},\sqrt{2} \right ) \sqrt{-3\,{x}^{2}+1}{\frac{1}{\sqrt{3\,{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*EllipticE(1/2*x*2^(1/2)*3^(1/2),2^(1/2))*(-3*x^2+1)^(1/2)*3^(1/2)/(3*x^2-1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{3 \, x^{2} - 1}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(3*x^2 - 1)/sqrt(-3*x^2 + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2}}{3 \, x^{2} - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2)/(3*x^2 - 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{3 x^{2} - 1}}{\sqrt{2 - 3 x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(3*x**2 - 1)/sqrt(2 - 3*x**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{3 \, x^{2} - 1}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(3*x^2 - 1)/sqrt(-3*x^2 + 2), x)