Optimal. Leaf size=191 \[ \frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}}-\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{d x^2-c}} \]
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Rubi [A] time = 0.37521, 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 \[ \frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}}-\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{d x^2-c}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-c + d*x^2]/Sqrt[a + b*x^2],x]
[Out]
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Rubi in Sympy [A] time = 75.2671, size = 162, normalized size = 0.85 \[ \frac{\sqrt{c} \sqrt{d} \sqrt{1 - \frac{d x^{2}}{c}} \sqrt{a + b x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{b \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{- c + d x^{2}}} - \frac{\sqrt{c} \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{1 - \frac{d x^{2}}{c}} \left (a d + b c\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a + b x^{2}} \sqrt{- c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2-c)**(1/2)/(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0669913, size = 90, normalized size = 0.47 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{d x^2-c} E\left (\sin ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c-d x^2}{c}}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-c + d*x^2]/Sqrt[a + b*x^2],x]
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Maple [A] time = 0.022, size = 109, normalized size = 0.6 \[{\frac{c}{-bd{x}^{4}-ad{x}^{2}+c{x}^{2}b+ac}\sqrt{d{x}^{2}-c}\sqrt{b{x}^{2}+a}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{-{\frac{d{x}^{2}-c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2-c)^(1/2)/(b*x^2+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} - c}}{\sqrt{b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 - c)/sqrt(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x^{2} - c}}{\sqrt{b x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 - c)/sqrt(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- c + d x^{2}}}{\sqrt{a + b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2-c)**(1/2)/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} - c}}{\sqrt{b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 - c)/sqrt(b*x^2 + a),x, algorithm="giac")
[Out]