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

Optimal. Leaf size=191 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{d \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}}-\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{b x^2-a} \sqrt{c+d x^2}} \]

[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]) - (Sqrt[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 = 0.382336, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{d \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}}-\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{b x^2-a} \sqrt{c+d x^2}} \]

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]) - (Sqrt[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 = 75.6895, size = 162, normalized size = 0.85 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{c + d x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{d \sqrt{1 + \frac{d x^{2}}{c}} \sqrt{- a + b x^{2}}} - \frac{\sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{1 + \frac{d x^{2}}{c}} \left (a d + b c\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{- a + b x^{2}} \sqrt{c + d x^{2}}} \]

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/sq
rt(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 = 0.0756661, size = 90, 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 = 0.022, size = 109, normalized size = 0.6 \[{\frac{a}{-bd{x}^{4}+ad{x}^{2}-c{x}^{2}b+ac}\sqrt{b{x}^{2}-a}\sqrt{d{x}^{2}+c}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

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]

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