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