3.649 \(\int \frac{1}{\sqrt{x} \sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=21 \[ \frac{2 F\left (\left .\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right |-1\right )}{\sqrt{a}} \]

[Out]

(2*EllipticF[ArcSin[Sqrt[a]*Sqrt[x]], -1])/Sqrt[a]

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Rubi [A]  time = 0.0329877, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 F\left (\left .\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right |-1\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

(2*EllipticF[ArcSin[Sqrt[a]*Sqrt[x]], -1])/Sqrt[a]

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Rubi in Sympy [A]  time = 5.3998, size = 20, normalized size = 0.95 \[ \frac{2 F\left (\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}\middle | -1\right )}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(1/2)/(-a**2*x**2+1)**(1/2),x)

[Out]

2*elliptic_f(asin(sqrt(a)*sqrt(x)), -1)/sqrt(a)

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Mathematica [C]  time = 0.125551, size = 65, normalized size = 3.1 \[ -\frac{2 i \sqrt{-\frac{1}{a}} a x \sqrt{1-\frac{1}{a^2 x^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\frac{1}{a}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*I)*Sqrt[-a^(-1)]*a*Sqrt[1 - 1/(a^2*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[-a^(-1)
]/Sqrt[x]], -1])/Sqrt[1 - a^2*x^2]

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Maple [B]  time = 0.073, size = 66, normalized size = 3.1 \[ -{\frac{1}{a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax+1}\sqrt{-2\,ax+2}\sqrt{-ax}{\it EllipticF} \left ( \sqrt{ax+1},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/2)/(-a^2*x^2+1)^(1/2),x)

[Out]

-1/x^(1/2)*(-a^2*x^2+1)^(1/2)*(a*x+1)^(1/2)*(-2*a*x+2)^(1/2)*(-a*x)^(1/2)*Ellipt
icF((a*x+1)^(1/2),1/2*2^(1/2))/a/(a^2*x^2-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-a^{2} x^{2} + 1} \sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-a^2*x^2 + 1)*sqrt(x)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-a^{2} x^{2} + 1} \sqrt{x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-a^2*x^2 + 1)*sqrt(x)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(-a^2*x^2 + 1)*sqrt(x)), x)

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Sympy [A]  time = 2.30182, size = 36, normalized size = 1.71 \[ \frac{\sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(1/2)/(-a**2*x**2+1)**(1/2),x)

[Out]

sqrt(x)*gamma(1/4)*hyper((1/4, 1/2), (5/4,), a**2*x**2*exp_polar(2*I*pi))/(2*gam
ma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-a^{2} x^{2} + 1} \sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-a^2*x^2 + 1)*sqrt(x)),x, algorithm="giac")

[Out]

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