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

Optimal. Leaf size=91 \[ \frac{(c+4 d) \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{c+d x^2}}-\frac{\sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{\frac{d x^2}{c}+1}} \]

[Out]

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

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Rubi [A]  time = 0.185635, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{(c+4 d) \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{c+d x^2}}-\frac{\sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{\frac{d x^2}{c}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.1494, size = 75, normalized size = 0.82 \[ \frac{\sqrt{1 + \frac{d x^{2}}{c}} \left (c + 4 d\right ) F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | - \frac{4 d}{c}\right )}{d \sqrt{c + d x^{2}}} - \frac{\sqrt{c + d x^{2}} E\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | - \frac{4 d}{c}\right )}{d \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0568146, size = 60, normalized size = 0.66 \[ \frac{2 \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c}{4 d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.033, size = 78, normalized size = 0.9 \[{\frac{1}{d} \left ( c{\it EllipticF} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ) +4\,{\it EllipticF} \left ( x/2,2\,\sqrt{-{\frac{d}{c}}} \right ) d-c{\it EllipticE} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ) \right ) \sqrt{{\frac{d{x}^{2}+c}{c}}}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*EllipticF(1/2*x,2*(-d/c)^(1/2))+4*EllipticF(1/2*x,2*(-d/c)^(1/2))*d-c*Ellipti
cE(1/2*x,2*(-d/c)^(1/2)))*((d*x^2+c)/c)^(1/2)/(d*x^2+c)^(1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(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. \[ \int \frac{\sqrt{-x^{2} + 4}}{\sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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