Optimal. Leaf size=189 \[ \frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}}-\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1}} \]
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Rubi [A] time = 0.391468, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}}-\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1}} \]
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 = 76.1481, size = 162, normalized size = 0.86 \[ - \frac{\sqrt{a} \sqrt{b} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{c + d x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{d \sqrt{1 + \frac{d x^{2}}{c}} \sqrt{a - b x^{2}}} + \frac{\sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{1 + \frac{d x^{2}}{c}} \left (a d + b c\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a - b x^{2}} \sqrt{c + d x^{2}}} \]
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 = 0.0742127, size = 89, normalized size = 0.47 \[ \frac{\sqrt{a-b x^2} \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{b c}{a d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a-b x^2}{a}} \sqrt{c+d x^2}} \]
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 = 0.018, size = 164, normalized size = 0.9 \[{\frac{1}{ \left ( bd{x}^{4}-ad{x}^{2}+c{x}^{2}b-ac \right ) d} \left ( -a{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) d-bc{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) +bc{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) \right ) \sqrt{-b{x}^{2}+a}\sqrt{d{x}^{2}+c}\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\frac{1}{\sqrt{{\frac{b}{a}}}}}} \]
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]