Optimal. Leaf size=232 \[ -\frac{d \sqrt{a+\frac{b}{x^2}}}{c x \sqrt{c+\frac{d}{x^2}}}+\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c}-\frac{b \sqrt{c} \sqrt{a+\frac{b}{x^2}} F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}+\frac{\sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]
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Rubi [A] time = 0.587952, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304 \[ -\frac{d \sqrt{a+\frac{b}{x^2}}}{c x \sqrt{c+\frac{d}{x^2}}}+\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c}-\frac{b \sqrt{c} \sqrt{a+\frac{b}{x^2}} F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}+\frac{\sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]
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 = 56.6809, size = 199, normalized size = 0.86 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{c + \frac{d}{x^{2}}} E\left (\operatorname{atan}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}\middle | - \frac{a d}{b c} + 1\right )}{c \sqrt{\frac{a \left (c + \frac{d}{x^{2}}\right )}{c \left (a + \frac{b}{x^{2}}\right )}} \sqrt{a + \frac{b}{x^{2}}}} - \frac{b \sqrt{c + \frac{d}{x^{2}}}}{c x \sqrt{a + \frac{b}{x^{2}}}} + \frac{x \sqrt{a + \frac{b}{x^{2}}} \sqrt{c + \frac{d}{x^{2}}}}{c} - \frac{b \sqrt{c} \sqrt{a + \frac{b}{x^{2}}} F\left (\operatorname{atan}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}\middle | 1 - \frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{\frac{c \left (a + \frac{b}{x^{2}}\right )}{a \left (c + \frac{d}{x^{2}}\right )}} \sqrt{c + \frac{d}{x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)**(1/2)/(c+d/x**2)**(1/2),x)
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Mathematica [A] time = 0.101855, size = 86, normalized size = 0.37 \[ \frac{\sqrt{a+\frac{b}{x^2}} \sqrt{\frac{c x^2+d}{d}} E\left (\sin ^{-1}\left (\sqrt{-\frac{c}{d}} x\right )|\frac{a d}{b c}\right )}{\sqrt{-\frac{c}{d}} \sqrt{\frac{a x^2+b}{b}} \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x^2]/Sqrt[c + d/x^2],x]
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Maple [A] time = 0.029, size = 94, normalized size = 0.4 \[{\frac{b}{a{x}^{2}+b}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}{\it EllipticE} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) \sqrt{{\frac{a{x}^{2}+b}{b}}}\sqrt{{\frac{c{x}^{2}+d}{d}}}{\frac{1}{\sqrt{-{\frac{c}{d}}}}}{\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)^(1/2)/(c+d/x^2)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{c + \frac{d}{x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^2)/sqrt(c + d/x^2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^2)/sqrt(c + d/x^2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{c + \frac{d}{x^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)**(1/2)/(c+d/x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{c + \frac{d}{x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^2)/sqrt(c + d/x^2),x, algorithm="giac")
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