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

Optimal. Leaf size=198 \[ -\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{b x^2-a} \sqrt{-c-d x^2}}-\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{d \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}} \]

[Out]

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

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Rubi [A]  time = 0.39615, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{b x^2-a} \sqrt{-c-d x^2}}-\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{d \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 93.126, size = 167, normalized size = 0.84 \[ - \frac{\sqrt{a} \sqrt{b} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{- c - d x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{d \sqrt{1 + \frac{d x^{2}}{c}} \sqrt{- a + b x^{2}}} - \frac{\sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{1 + \frac{d x^{2}}{c}} \left (a d + b c\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{- a + b x^{2}} \sqrt{- c - d x^{2}}} \]

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

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Mathematica [A]  time = 0.0649757, size = 93, normalized size = 0.47 \[ \frac{\sqrt{b x^2-a} \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{b c}{a d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a-b x^2}{a}} \sqrt{-c-d x^2}} \]

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 [A]  time = 0.017, size = 167, normalized size = 0.8 \[{\frac{1}{ \left ( bd{x}^{4}-ad{x}^{2}+c{x}^{2}b-ac \right ) d}\sqrt{b{x}^{2}-a}\sqrt{-d{x}^{2}-c}\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}} \left ( a{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) d+bc{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) -bc{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{a}}}}}} \]

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*Ell
ipticF(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)