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

Optimal. Leaf size=89 \[ \frac{\sqrt{c} \sqrt{b x^2-a} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2}} \]

[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 = 0.159725, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{c} \sqrt{b x^2-a} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2}} \]

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 = 38.938, size = 73, normalized size = 0.82 \[ \frac{\sqrt{c} \sqrt{1 - \frac{d x^{2}}{c}} \sqrt{- a + b x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | \frac{b c}{a d}\right )}{\sqrt{d} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{c - d x^{2}}} \]

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 = 0.0718343, size = 89, normalized size = 1. \[ \frac{\sqrt{b x^2-a} \sqrt{\frac{c-d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{b c}{a d}\right )}{\sqrt{\frac{d}{c}} \sqrt{\frac{a-b x^2}{a}} \sqrt{c-d x^2}} \]

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 [B]  time = 0.025, size = 165, normalized size = 1.9 \[{\frac{1}{ \left ( bd{x}^{4}-ad{x}^{2}-c{x}^{2}b+ac \right ) d}\sqrt{b{x}^{2}-a}\sqrt{-d{x}^{2}+c}\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{-{\frac{d{x}^{2}-c}{c}}} \left ( a{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) d-bc{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) +bc{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{a}}}}}} \]

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

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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)