3.193 \(\int \frac {\sqrt {4-x^2}}{\sqrt {2+3 x^2}} \, dx\)

Optimal. Leaf size=35 \[ \frac {7}{3} \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {x}{2}\right ),-6\right )-\frac {1}{3} \sqrt {2} E\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-6\right ) \]

[Out]

-1/3*EllipticE(1/2*x,I*6^(1/2))*2^(1/2)+7/3*EllipticF(1/2*x,I*6^(1/2))*2^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {423, 424, 419} \[ \frac {7}{3} \sqrt {2} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-6\right )-\frac {1}{3} \sqrt {2} E\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-6\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[4 - x^2]/Sqrt[2 + 3*x^2],x]

[Out]

-(Sqrt[2]*EllipticE[ArcSin[x/2], -6])/3 + (7*Sqrt[2]*EllipticF[ArcSin[x/2], -6])/3

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)])

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {4-x^2}}{\sqrt {2+3 x^2}} \, dx &=-\left (\frac {1}{3} \int \frac {\sqrt {2+3 x^2}}{\sqrt {4-x^2}} \, dx\right )+\frac {14}{3} \int \frac {1}{\sqrt {4-x^2} \sqrt {2+3 x^2}} \, dx\\ &=-\frac {1}{3} \sqrt {2} E\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-6\right )+\frac {7}{3} \sqrt {2} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-6\right )\\ \end {align*}

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Mathematica [C]  time = 0.00, size = 27, normalized size = 0.77 \[ -\frac {2 i E\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[4 - x^2]/Sqrt[2 + 3*x^2],x]

[Out]

((-2*I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], -1/6])/Sqrt[3]

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fricas [F]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 4}}{\sqrt {3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+4)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(-x^2 + 4)/sqrt(3*x^2 + 2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 4}}{\sqrt {3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+4)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-x^2 + 4)/sqrt(3*x^2 + 2), x)

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maple [A]  time = 0.03, size = 31, normalized size = 0.89 \[ \frac {\left (-\EllipticE \left (\frac {x}{2}, i \sqrt {6}\right )+7 \EllipticF \left (\frac {x}{2}, i \sqrt {6}\right )\right ) \sqrt {2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(7*EllipticF(1/2*x,I*6^(1/2))-EllipticE(1/2*x,I*6^(1/2)))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 4}}{\sqrt {3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+4)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^2 + 4)/sqrt(3*x^2 + 2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {4-x^2}}{\sqrt {3\,x^2+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (x - 2\right ) \left (x + 2\right )}}{\sqrt {3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+4)**(1/2)/(3*x**2+2)**(1/2),x)

[Out]

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

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