Optimal. Leaf size=66 \[ \frac{B \sqrt{b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}} \]
[Out]
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Rubi [A] time = 0.224472, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{B \sqrt{b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]
[Out]
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Rubi in Sympy [A] time = 17.6371, size = 56, normalized size = 0.85 \[ \frac{B \sqrt{b x^{2} + c x^{4}}}{2 c} + \frac{\left (2 A c - B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{2 c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.084591, size = 84, normalized size = 1.27 \[ \frac{x \left (\sqrt{b+c x^2} (2 A c-b B) \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )+B \sqrt{c} x \left (b+c x^2\right )\right )}{2 c^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]
[Out]
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Maple [A] time = 0.009, size = 88, normalized size = 1.3 \[{\frac{x}{2}\sqrt{c{x}^{2}+b} \left ( Bx\sqrt{c{x}^{2}+b}{c}^{{\frac{3}{2}}}+2\,A\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){c}^{2}-Bb\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ) c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(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)*x/sqrt(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233946, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{c x^{4} + b x^{2}} B c -{\left (B b - 2 \, A c\right )} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{4} + b x^{2}} c\right )}{4 \, c^{2}}, \frac{\sqrt{c x^{4} + b x^{2}} B c +{\left (B b - 2 \, A c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right )}{2 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/sqrt(c*x^4 + b*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (A + B x^{2}\right )}{\sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.242774, size = 90, normalized size = 1.36 \[ \frac{\sqrt{c x^{4} + b x^{2}} B}{2 \, c} + \frac{{\left (B b - 2 \, A c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2}}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/sqrt(c*x^4 + b*x^2),x, algorithm="giac")
[Out]