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

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

[Out]

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

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Rubi [A]  time = 0.161963, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{\sqrt{c} \sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.1882, size = 76, normalized size = 0.84 \[ \frac{\sqrt{c} \sqrt{1 - \frac{d x^{2}}{c}} \sqrt{- a - b x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{\sqrt{d} \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{- c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0720752, size = 91, normalized size = 1. \[ \frac{\sqrt{-a-b x^2} \sqrt{\frac{c-d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|-\frac{b c}{a d}\right )}{\sqrt{\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{d x^2-c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 110, normalized size = 1.2 \[{\frac{a}{bd{x}^{4}+ad{x}^{2}-c{x}^{2}b-ac}\sqrt{-b{x}^{2}-a}\sqrt{d{x}^{2}-c}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}}{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ){\frac{1}{\sqrt{{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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