3.3.27 \(\int \frac {1}{\sqrt {2-5 x^2} \sqrt {1-x^2}} \, dx\) [227]

Optimal. Leaf size=12 \[ \frac {F\left (\sin ^{-1}(x)|\frac {5}{2}\right )}{\sqrt {2}} \]

[Out]

1/2*EllipticF(x,1/2*10^(1/2))*2^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 12, 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 = {430} \begin {gather*} \frac {F\left (\text {ArcSin}(x)\left |\frac {5}{2}\right .\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[2 - 5*x^2]*Sqrt[1 - x^2]),x]

[Out]

EllipticF[ArcSin[x], 5/2]/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])

Rubi steps

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

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Mathematica [A]
time = 0.22, size = 12, normalized size = 1.00 \begin {gather*} \frac {F\left (\sin ^{-1}(x)|\frac {5}{2}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[2 - 5*x^2]*Sqrt[1 - x^2]),x]

[Out]

EllipticF[ArcSin[x], 5/2]/Sqrt[2]

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Maple [A]
time = 0.10, size = 13, normalized size = 1.08

method	result	size
default	\(\frac {\EllipticF \left (x , \frac {\sqrt {10}}{2}\right ) \sqrt {2}}{2}\)	\(13\)
elliptic	\(\frac {\sqrt {\left (5 x^{2}-2\right ) \left (x^{2}-1\right )}\, \sqrt {-10 x^{2}+4}\, \EllipticF \left (x , \frac {\sqrt {10}}{2}\right )}{2 \sqrt {-5 x^{2}+2}\, \sqrt {5 x^{4}-7 x^{2}+2}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-5*x^2+2)^(1/2)/(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*EllipticF(x,1/2*10^(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(1/(-5*x^2+2)^(1/2)/(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^2 + 1)*sqrt(-5*x^2 + 2)), x)

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Fricas [A]
time = 0.21, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \sqrt {2} {\rm ellipticF}\left (x, \frac {5}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5*x^2+2)^(1/2)/(-x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*ellipticF(x, 5/2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).
time = 1.34, size = 34, normalized size = 2.83 \begin {gather*} \begin {cases} \frac {\sqrt {5} F\left (\operatorname {asin}{\left (\frac {\sqrt {10} x}{2} \right )}\middle | \frac {2}{5}\right )}{5} & \text {for}\: x > - \frac {\sqrt {10}}{5} \wedge x < \frac {\sqrt {10}}{5} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5*x**2+2)**(1/2)/(-x**2+1)**(1/2),x)

[Out]

Piecewise((sqrt(5)*elliptic_f(asin(sqrt(10)*x/2), 2/5)/5, (x > -sqrt(10)/5) & (x < sqrt(10)/5)))

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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(1/(-5*x^2+2)^(1/2)/(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^2 + 1)*sqrt(-5*x^2 + 2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int \frac {1}{\sqrt {1-x^2}\,\sqrt {2-5\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x^2)^(1/2)*(2 - 5*x^2)^(1/2)),x)

[Out]

int(1/((1 - x^2)^(1/2)*(2 - 5*x^2)^(1/2)), x)

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