3.2.92 \(\int \frac {\sqrt {1-x^2}}{\sqrt {2+3 x^2}} \, dx\) [192]

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

[Out]

5/6*EllipticF(x,1/2*I*6^(1/2))*2^(1/2)-1/3*EllipticE(x,1/2*I*6^(1/2))*2^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 31, 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 = {434, 435, 430} \begin {gather*} \frac {5 F\left (\text {ArcSin}(x)\left |-\frac {3}{2}\right .\right )}{3 \sqrt {2}}-\frac {1}{3} \sqrt {2} E\left (\text {ArcSin}(x)\left |-\frac {3}{2}\right .\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/Sqrt[2 + 3*x^2],x]

[Out]

-1/3*(Sqrt[2]*EllipticE[ArcSin[x], -3/2]) + (5*EllipticF[ArcSin[x], -3/2])/(3*Sqrt[2])

Rule 430

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

Rule 434

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 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], 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 &=-\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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.29, size = 27, normalized size = 0.87 \begin {gather*} -\frac {i E\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {3}} \end {gather*}

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 = 0.09, size = 27, normalized size = 0.87

method	result	size
default	\(\frac {\left (5 \EllipticF \left (x , \frac {i \sqrt {6}}{2}\right )-2 \EllipticE \left (x , \frac {i \sqrt {6}}{2}\right )\right ) \sqrt {2}}{6}\)	\(27\)
elliptic	\(\frac {\sqrt {-\left (3 x^{2}+2\right ) \left (x^{2}-1\right )}\, \left (\frac {\sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+4}\, \EllipticF \left (x , \frac {i \sqrt {6}}{2}\right )}{2 \sqrt {-3 x^{4}+x^{2}+2}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\EllipticF \left (x , \frac {i \sqrt {6}}{2}\right )-\EllipticE \left (x , \frac {i \sqrt {6}}{2}\right )\right )}{3 \sqrt {-3 x^{4}+x^{2}+2}}\right )}{\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+2}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, 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((-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 [A]
time = 0.23, size = 23, normalized size = 0.74 \begin {gather*} \frac {\sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 1}}{3 \, x} \end {gather*}

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]

1/3*sqrt(3*x^2 + 2)*sqrt(-x^2 + 1)/x

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

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 = 0.00, 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((-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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {1-x^2}}{\sqrt {3\,x^2+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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