3.214 \(\int \frac{1}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{\sqrt{c} \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}} \]

[Out]

(Sqrt[c]*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sq
rt[c]], (b*c)/(a*d)])/(Sqrt[d]*Sqrt[a - b*x^2]*Sqrt[c - d*x^2])

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Rubi [A]  time = 0.171007, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{\sqrt{c} \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sq
rt[c]], (b*c)/(a*d)])/(Sqrt[d]*Sqrt[a - b*x^2]*Sqrt[c - d*x^2])

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Rubi in Sympy [A]  time = 54.7996, size = 73, normalized size = 0.83 \[ \frac{\sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{1 - \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | \frac{a d}{b c}\right )}{\sqrt{b} \sqrt{a - b x^{2}} \sqrt{c - d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0925442, size = 88, normalized size = 1. \[ \frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{\frac{c-d x^2}{c}} F\left (\sin ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{\sqrt{\frac{b}{a}} \sqrt{a-b x^2} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.041, size = 108, normalized size = 1.2 \[{\frac{1}{bd{x}^{4}-ad{x}^{2}-c{x}^{2}b+ac}{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) \sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-b{x}^{2}+a}\sqrt{-d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

EllipticF(x*(d/c)^(1/2),(b*c/a/d)^(1/2))*(-(b*x^2-a)/a)^(1/2)*(-(d*x^2-c)/c)^(1/
2)*(-b*x^2+a)^(1/2)*(-d*x^2+c)^(1/2)/(d/c)^(1/2)/(b*d*x^4-a*d*x^2-b*c*x^2+a*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a - b x^{2}} \sqrt{c - d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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