3.3.37 \(\int \frac {1}{\sqrt {2-5 x^2} \sqrt {1+x^2}} \, dx\) [237]

Optimal. Leaf size=20 \[ \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {5}{2}} x\right )|-\frac {2}{5}\right )}{\sqrt {5}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {430} \begin {gather*} \frac {F\left (\text {ArcSin}\left (\sqrt {\frac {5}{2}} x\right )|-\frac {2}{5}\right )}{\sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[Sqrt[5/2]*x], -2/5]/Sqrt[5]

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}\left (\sqrt {\frac {5}{2}} x\right )|-\frac {2}{5}\right )}{\sqrt {5}}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 20, normalized size = 1.00 \begin {gather*} \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {5}{2}} x\right )|-\frac {2}{5}\right )}{\sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[Sqrt[5/2]*x], -2/5]/Sqrt[5]

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Maple [A]
time = 0.09, size = 19, normalized size = 0.95

method	result	size
default	\(\frac {\EllipticF \left (\frac {x \sqrt {10}}{2}, \frac {i \sqrt {10}}{5}\right ) \sqrt {5}}{5}\)	\(19\)
elliptic	\(\frac {\sqrt {-\left (5 x^{2}-2\right ) \left (x^{2}+1\right )}\, \sqrt {10}\, \sqrt {-10 x^{2}+4}\, \EllipticF \left (\frac {x \sqrt {10}}{2}, \frac {i \sqrt {10}}{5}\right )}{10 \sqrt {-5 x^{2}+2}\, \sqrt {-5 x^{4}-3 x^{2}+2}}\)	\(67\)

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]

1/5*EllipticF(1/2*x*10^(1/2),1/5*I*10^(1/2))*5^(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 = 0.15, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{5} \, \sqrt {5} {\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]

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

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Sympy [A]
time = 1.50, size = 36, normalized size = 1.80 \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 = -0.05 \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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