Optimal. Leaf size=34 \[ \frac{(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac{A \log (x)}{b} \]
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Rubi [A] time = 0.104874, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac{A \log (x)}{b} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x^2))/(b*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 14.5494, size = 29, normalized size = 0.85 \[ \frac{A \log{\left (x^{2} \right )}}{2 b} - \frac{\left (A c - B b\right ) \log{\left (b + c x^{2} \right )}}{2 b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x**2+A)/(c*x**4+b*x**2),x)
[Out]
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Mathematica [A] time = 0.0205189, size = 34, normalized size = 1. \[ \frac{(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac{A \log (x)}{b} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x^2))/(b*x^2 + c*x^4),x]
[Out]
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Maple [A] time = 0.007, size = 37, normalized size = 1.1 \[{\frac{A\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( c{x}^{2}+b \right ) A}{2\,b}}+{\frac{\ln \left ( c{x}^{2}+b \right ) B}{2\,c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x^2+A)/(c*x^4+b*x^2),x)
[Out]
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Maxima [A] time = 1.37236, size = 47, normalized size = 1.38 \[ \frac{A \log \left (x^{2}\right )}{2 \, b} + \frac{{\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210755, size = 43, normalized size = 1.26 \[ \frac{2 \, A c \log \left (x\right ) +{\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(c*x^4 + b*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.10029, size = 26, normalized size = 0.76 \[ \frac{A \log{\left (x \right )}}{b} + \frac{\left (- A c + B b\right ) \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x**2+A)/(c*x**4+b*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.211408, size = 46, normalized size = 1.35 \[ \frac{A{\rm ln}\left ({\left | x \right |}\right )}{b} + \frac{{\left (B b - A c\right )}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(c*x^4 + b*x^2),x, algorithm="giac")
[Out]