3.3.96 \(\int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx\) [296]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {436} \begin {gather*} -\frac {E\left (\left .\text {ArcCos}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3}} \end {gather*}

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 436

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(-Sqrt[a - b*(c/d)]/(Sqrt[c]*Rt[-d/
c, 2]))*EllipticE[ArcCos[Rt[-d/c, 2]*x], b*(c/(b*c - a*d))], 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*}

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Mathematica [A]
time = 0.32, size = 35, normalized size = 1.84 \begin {gather*} \frac {\sqrt {-1+3 x^2} E\left (\left .\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3-9 x^2}} \end {gather*}

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 = 0.11, size = 37, normalized size = 1.95

method	result	size
default	\(-\frac {\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}, \sqrt {2}\right ) \sqrt {-3 x^{2}+1}\, \sqrt {3}}{3 \sqrt {3 x^{2}-1}}\)	\(37\)
elliptic	\(\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \left (-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \EllipticF \left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )}{6 \sqrt {-9 x^{4}+9 x^{2}-2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \left (\EllipticF \left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )-\EllipticE \left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )\right )}{6 \sqrt {-9 x^{4}+9 x^{2}-2}}\right )}{\sqrt {3 x^{2}-1}\, \sqrt {-3 x^{2}+2}}\)	\(146\)

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,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [A]
time = 0.14, size = 23, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {3 \, x^{2} - 1} \sqrt {-3 \, x^{2} + 2}}{3 \, x} \end {gather*}

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]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {3 x^{2} - 1}}{\sqrt {2 - 3 x^{2}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\sqrt {3\,x^2-1}}{\sqrt {2-3\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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