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

Optimal. Leaf size=20 \[ \frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )}{\sqrt {3}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 20, 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 = {424} \[ \frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/Sqrt[2 - 3*x^2],x]

[Out]

EllipticE[ArcSin[Sqrt[3/2]*x], 2/3]/Sqrt[3]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (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, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \[ \frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticE[ArcSin[Sqrt[3/2]*x], 2/3]/Sqrt[3]

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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}{3 \, x^{2} - 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 23, normalized size = 1.15 \[ \frac {\sqrt {2}\, \left (2 \EllipticE \left (x , \frac {\sqrt {6}}{2}\right )+\EllipticF \left (x , \frac {\sqrt {6}}{2}\right )\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*2^(1/2)*(EllipticF(x,1/2*6^(1/2))+2*EllipticE(x,1/2*6^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\sqrt {1-x^2}}{\sqrt {2-3\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.94, size = 34, normalized size = 1.70 \[ \begin {cases} \frac {\sqrt {3} E\left (\operatorname {asin}{\left (\frac {\sqrt {6} x}{2} \right )}\middle | \frac {2}{3}\right )}{3} & \text {for}\: x > - \frac {\sqrt {6}}{3} \wedge x < \frac {\sqrt {6}}{3} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(3)*elliptic_e(asin(sqrt(6)*x/2), 2/3)/3, (x > -sqrt(6)/3) & (x < sqrt(6)/3)))

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