Optimal. Leaf size=55 \[ \frac{B \sqrt{b x^2+c x^4}}{c x}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{b}} \]
[Out]
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Rubi [A] time = 0.0601968, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{B \sqrt{b x^2+c x^4}}{c x}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/Sqrt[b*x^2 + c*x^4],x]
[Out]
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Rubi in Sympy [A] time = 8.12021, size = 46, normalized size = 0.84 \[ - \frac{A \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{\sqrt{b}} + \frac{B \sqrt{b x^{2} + c x^{4}}}{c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.095474, size = 91, normalized size = 1.65 \[ \frac{x \left (A c \log (x) \sqrt{b+c x^2}-A c \sqrt{b+c x^2} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+\sqrt{b} B \left (b+c x^2\right )\right )}{\sqrt{b} c \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/Sqrt[b*x^2 + c*x^4],x]
[Out]
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Maple [A] time = 0.014, size = 72, normalized size = 1.3 \[ -{\frac{x}{c}\sqrt{c{x}^{2}+b} \left ( A\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) c-B\sqrt{c{x}^{2}+b}\sqrt{b} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/(c*x^4+b*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((B*x^2 + A)/sqrt(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235248, size = 1, normalized size = 0.02 \[ \left [\frac{A \sqrt{b} c x \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}} B b}{2 \, b c x}, \frac{A \sqrt{-b} c x \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}} B b}{b c x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/sqrt(c*x^4 + b*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x^{2}}{\sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.238072, size = 81, normalized size = 1.47 \[ \frac{A{\rm ln}\left ({\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2}\right )}{2 \, \sqrt{b}} - \frac{2 \, B \sqrt{b}}{{\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2} - c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/sqrt(c*x^4 + b*x^2),x, algorithm="giac")
[Out]