3.2.94 \(\int \frac {\sqrt {1-4 x^2}}{\sqrt {2+3 x^2}} \, dx\) [194]

Optimal. Leaf size=35 \[ -\frac {2}{3} \sqrt {2} E\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )+\frac {11 F\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{6 \sqrt {2}} \]

[Out]

11/12*EllipticF(2*x,1/4*I*6^(1/2))*2^(1/2)-2/3*EllipticE(2*x,1/4*I*6^(1/2))*2^(1/2)

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Rubi [A]
time = 0.01, 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 = {434, 435, 430} \begin {gather*} \frac {11 F\left (\text {ArcSin}(2 x)\left |-\frac {3}{8}\right .\right )}{6 \sqrt {2}}-\frac {2}{3} \sqrt {2} E\left (\text {ArcSin}(2 x)\left |-\frac {3}{8}\right .\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 4*x^2]/Sqrt[2 + 3*x^2],x]

[Out]

(-2*Sqrt[2]*EllipticE[ArcSin[2*x], -3/8])/3 + (11*EllipticF[ArcSin[2*x], -3/8])/(6*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 {1-4 x^2}}{\sqrt {2+3 x^2}} \, dx &=-\left (\frac {4}{3} \int \frac {\sqrt {2+3 x^2}}{\sqrt {1-4 x^2}} \, dx\right )+\frac {11}{3} \int \frac {1}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx\\ &=-\frac {2}{3} \sqrt {2} E\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )+\frac {11 F\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{6 \sqrt {2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.30, size = 27, normalized size = 0.77 \begin {gather*} -\frac {i E\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 4*x^2]/Sqrt[2 + 3*x^2],x]

[Out]

((-I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], -8/3])/Sqrt[3]

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Maple [A]
time = 0.10, size = 31, normalized size = 0.89

method	result	size
default	\(\frac {\left (11 \EllipticF \left (2 x , \frac {i \sqrt {6}}{4}\right )-8 \EllipticE \left (2 x , \frac {i \sqrt {6}}{4}\right )\right ) \sqrt {2}}{12}\)	\(31\)
elliptic	\(\frac {\sqrt {-\left (3 x^{2}+2\right ) \left (4 x^{2}-1\right )}\, \left (\frac {\sqrt {-4 x^{2}+1}\, \sqrt {6 x^{2}+4}\, \EllipticF \left (2 x , \frac {i \sqrt {6}}{4}\right )}{4 \sqrt {-12 x^{4}-5 x^{2}+2}}+\frac {2 \sqrt {-4 x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\EllipticF \left (2 x , \frac {i \sqrt {6}}{4}\right )-\EllipticE \left (2 x , \frac {i \sqrt {6}}{4}\right )\right )}{3 \sqrt {-12 x^{4}-5 x^{2}+2}}\right )}{\sqrt {-4 x^{2}+1}\, \sqrt {3 x^{2}+2}}\)	\(140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(11*EllipticF(2*x,1/4*I*6^(1/2))-8*EllipticE(2*x,1/4*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((-4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.15, size = 23, normalized size = 0.66 \begin {gather*} \frac {\sqrt {3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3*x^2 + 2)*sqrt(-4*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 (2 x - 1\right ) \left (2 x + 1\right )}}{\sqrt {3 x^{2} + 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(sqrt(-4*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 {1-4\,x^2}}{\sqrt {3\,x^2+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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