3.288 \(\int \frac{1}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{\sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]

[Out]

(Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/Sqrt[c + d*x^2]

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Rubi [A]  time = 0.07319, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/Sqrt[c + d*x^2]

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Rubi in Sympy [A]  time = 15.8944, size = 32, normalized size = 0.82 \[ \frac{\sqrt{1 + \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | - \frac{4 d}{c}\right )}{\sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

sqrt(1 + d*x**2/c)*elliptic_f(asin(x/2), -4*d/c)/sqrt(c + d*x**2)

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Mathematica [A]  time = 0.0560501, size = 40, normalized size = 1.03 \[ \frac{\sqrt{\frac{c+d x^2}{c}} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[(c + d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/Sqrt[c + d*x^2]

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Maple [A]  time = 0.032, size = 38, normalized size = 1. \[{1\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x)

[Out]

1/(d*x^2+c)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(1/2*x,2*(-d/c)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (x - 2\right ) \left (x + 2\right )} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(1/(sqrt(-(x - 2)*(x + 2))*sqrt(c + d*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)),x, algorithm="giac")

[Out]

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