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

Optimal. Leaf size=31 \[ \frac{5 \text{EllipticF}\left (\sin ^{-1}(x),-\frac{3}{2}\right )}{3 \sqrt{2}}-\frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|-\frac{3}{2}\right ) \]

[Out]

-(Sqrt[2]*EllipticE[ArcSin[x], -3/2])/3 + (5*EllipticF[ArcSin[x], -3/2])/(3*Sqrt[2])

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Rubi [A]  time = 0.0202918, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {423, 424, 419} \[ \frac{5 F\left (\sin ^{-1}(x)|-\frac{3}{2}\right )}{3 \sqrt{2}}-\frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|-\frac{3}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/Sqrt[2 + 3*x^2],x]

[Out]

-(Sqrt[2]*EllipticE[ArcSin[x], -3/2])/3 + (5*EllipticF[ArcSin[x], -3/2])/(3*Sqrt[2])

Rule 423

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[b/d, Int[Sqrt[c + d*x^2]/Sqrt[a + b
*x^2], x], x] - Dist[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x]
&& PosQ[d/c] && NegQ[b/a]

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]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{\sqrt{1-x^2}}{\sqrt{2+3 x^2}} \, dx &=-\left (\frac{1}{3} \int \frac{\sqrt{2+3 x^2}}{\sqrt{1-x^2}} \, dx\right )+\frac{5}{3} \int \frac{1}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx\\ &=-\frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|-\frac{3}{2}\right )+\frac{5 F\left (\sin ^{-1}(x)|-\frac{3}{2}\right )}{3 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0050336, size = 27, normalized size = 0.87 \[ -\frac{i E\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.014, size = 27, normalized size = 0.9 \begin{align*}{\frac{ \left ( 5\,{\it EllipticF} \left ( x,i/2\sqrt{6} \right ) -2\,{\it EllipticE} \left ( x,i/2\sqrt{6} \right ) \right ) \sqrt{2}}{6}} \end{align*}

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*(5*EllipticF(x,1/2*I*6^(1/2))-2*EllipticE(x,1/2*I*6^(1/2)))*2^(1/2)

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

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

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), x)

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

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]

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

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

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)