3.3.57 \(\int \frac {1}{\sqrt {2-5 x^2} \sqrt {-1-x^2}} \, dx\) [257]

Optimal. Leaf size=40 \[ \frac {\sqrt {1+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {5}{2}} x\right )|-\frac {2}{5}\right )}{\sqrt {5} \sqrt {-1-x^2}} \]

[Out]

1/5*EllipticF(1/2*x*10^(1/2),1/5*I*10^(1/2))*(x^2+1)^(1/2)*5^(1/2)/(-x^2-1)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {432, 430} \begin {gather*} \frac {\sqrt {x^2+1} F\left (\text {ArcSin}\left (\sqrt {\frac {5}{2}} x\right )|-\frac {2}{5}\right )}{\sqrt {5} \sqrt {-x^2-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[5/2]*x], -2/5])/(Sqrt[5]*Sqrt[-1 - x^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 432

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

Rubi steps

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

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Mathematica [A]
time = 0.21, size = 40, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {5}{2}} x\right )|-\frac {2}{5}\right )}{\sqrt {5} \sqrt {-1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.09, size = 34, normalized size = 0.85

method	result	size
default	\(\frac {i \EllipticF \left (i x , \frac {i \sqrt {10}}{2}\right ) \sqrt {2}\, \sqrt {-x^{2}-1}}{2 \sqrt {x^{2}+1}}\)	\(34\)
elliptic	\(-\frac {i \sqrt {\left (5 x^{2}-2\right ) \left (x^{2}+1\right )}\, \sqrt {x^{2}+1}\, \sqrt {-10 x^{2}+4}\, \EllipticF \left (i x , \frac {i \sqrt {10}}{2}\right )}{2 \sqrt {-5 x^{2}+2}\, \sqrt {-x^{2}-1}\, \sqrt {5 x^{4}+3 x^{2}-2}}\)	\(78\)

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*I*EllipticF(I*x,1/2*I*10^(1/2))*2^(1/2)/(x^2+1)^(1/2)*(-x^2-1)^(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 = 0.26, size = 22, normalized size = 0.55 \begin {gather*} -\frac {1}{10} \, \sqrt {5} \sqrt {2} \sqrt {-2} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {5} \sqrt {2} x, -\frac {2}{5}\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]

-1/10*sqrt(5)*sqrt(2)*sqrt(-2)*ellipticF(1/2*sqrt(5)*sqrt(2)*x, -2/5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 - 5 x^{2}} \sqrt {- x^{2} - 1}}\, dx \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]

Integral(1/(sqrt(2 - 5*x**2)*sqrt(-x**2 - 1)), 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(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 = -0.02 \begin {gather*} \int \frac {1}{\sqrt {-x^2-1}\,\sqrt {2-5\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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