Optimal. Leaf size=43 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{\sqrt{c}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d} \]
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Rubi [A] time = 0.12937, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{\sqrt{c}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)/(Sqrt[c + d/x^2]*x),x]
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Rubi in Sympy [A] time = 10.8749, size = 36, normalized size = 0.84 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{\sqrt{c}} - \frac{b \sqrt{c + \frac{d}{x^{2}}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)/x/(c+d/x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.063596, size = 76, normalized size = 1.77 \[ \frac{a d x \sqrt{c x^2+d} \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )-b \sqrt{c} \left (c x^2+d\right )}{\sqrt{c} d x^2 \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x),x]
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Maple [A] time = 0.016, size = 70, normalized size = 1.6 \[ -{\frac{1}{d{x}^{2}}\sqrt{c{x}^{2}+d} \left ( -a\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) xd+b\sqrt{c{x}^{2}+d}\sqrt{c} \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{\frac{1}{\sqrt{c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)/x/(c+d/x^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(sqrt(c + d/x^2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227924, size = 1, normalized size = 0.02 \[ \left [\frac{a \sqrt{c} d \log \left (-2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) - 2 \, b c \sqrt{\frac{c x^{2} + d}{x^{2}}}}{2 \, c d}, -\frac{a \sqrt{-c} d \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) + b c \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(sqrt(c + d/x^2)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.85985, size = 138, normalized size = 3.21 \[ - a \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{c}} \sqrt{c + \frac{d}{x^{2}}}} \right )}}{c \sqrt{- \frac{1}{c}}} & \text{for}\: - \frac{1}{c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{c + \frac{d}{x^{2}}} \sqrt{\frac{1}{c}}} \right )}}{c \sqrt{\frac{1}{c}}} & \text{for}\: - \frac{1}{c} < 0 \wedge \frac{1}{c} < \frac{1}{c + \frac{d}{x^{2}}} \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{c + \frac{d}{x^{2}}} \sqrt{\frac{1}{c}}} \right )}}{c \sqrt{\frac{1}{c}}} & \text{for}\: \frac{1}{c} > \frac{1}{c + \frac{d}{x^{2}}} \wedge - \frac{1}{c} < 0 \end{cases}\right ) - \frac{b \sqrt{c + \frac{d}{x^{2}}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)/x/(c+d/x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{\sqrt{c + \frac{d}{x^{2}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(sqrt(c + d/x^2)*x),x, algorithm="giac")
[Out]