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

Optimal. Leaf size=90 \[ \frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1}} \]

[Out]

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

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Rubi [A]  time = 0.158979, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.1347, size = 76, normalized size = 0.84 \[ \frac{\sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{- c - d x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{\sqrt{b} \sqrt{1 + \frac{d x^{2}}{c}} \sqrt{a - b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0685608, size = 90, normalized size = 1. \[ \frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{\sqrt{\frac{b}{a}} \sqrt{a-b x^2} \sqrt{\frac{c+d x^2}{c}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.025, size = 171, normalized size = 1.9 \[{\frac{1}{ \left ( bd{x}^{4}-ad{x}^{2}+c{x}^{2}b-ac \right ) b} \left ( -ad{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) -c{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) b+ad{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) \right ) \sqrt{-d{x}^{2}-c}\sqrt{-b{x}^{2}+a}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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