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

Optimal. Leaf size=87 \[ \frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}} \]

[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 = 0.15265, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}} \]

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 = 29.8198, size = 75, normalized size = 0.86 \[ \frac{\sqrt{c} \sqrt{1 - \frac{d x^{2}}{c}} \sqrt{a + b x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{\sqrt{d} \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{c - d x^{2}}} \]

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 = 0.0940049, size = 87, normalized size = 1. \[ \frac{\sqrt{a+b x^2} \sqrt{\frac{c-d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|-\frac{b c}{a d}\right )}{\sqrt{\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{c-d x^2}} \]

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 = 106, 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)