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