Optimal. Leaf size=41 \[ \frac{(A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]
[Out]
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Rubi [A] time = 0.0735945, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(b*x^2 - c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 11.6369, size = 34, normalized size = 0.83 \[ - \frac{A}{b x} + \frac{\left (A c + B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{b^{\frac{3}{2}} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/(-c*x**4+b*x**2),x)
[Out]
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Mathematica [A] time = 0.0416579, size = 41, normalized size = 1. \[ \frac{(A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(b*x^2 - c*x^4),x]
[Out]
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Maple [A] time = 0.009, size = 39, normalized size = 1. \[ -{\frac{-Ac-Bb}{b}{\it Artanh} \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{A}{bx}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/(-c*x^4+b*x^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(-(B*x^2 + A)/(c*x^4 - b*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219043, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (B b + A c\right )} x \log \left (\frac{2 \, b c x +{\left (c x^{2} + b\right )} \sqrt{b c}}{c x^{2} - b}\right ) - 2 \, \sqrt{b c} A}{2 \, \sqrt{b c} b x}, \frac{{\left (B b + A c\right )} x \arctan \left (\frac{\sqrt{-b c} x}{b}\right ) - \sqrt{-b c} A}{\sqrt{-b c} b x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(B*x^2 + A)/(c*x^4 - b*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.893399, size = 75, normalized size = 1.83 \[ - \frac{A}{b x} - \frac{\sqrt{\frac{1}{b^{3} c}} \left (A c + B b\right ) \log{\left (- b^{2} \sqrt{\frac{1}{b^{3} c}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{b^{3} c}} \left (A c + B b\right ) \log{\left (b^{2} \sqrt{\frac{1}{b^{3} c}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/(-c*x**4+b*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.208797, size = 51, normalized size = 1.24 \[ -\frac{{\left (B b + A c\right )} \arctan \left (\frac{c x}{\sqrt{-b c}}\right )}{\sqrt{-b c} b} - \frac{A}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(B*x^2 + A)/(c*x^4 - b*x^2),x, algorithm="giac")
[Out]