Optimal. Leaf size=15 \[ \frac{1}{2} \log \left (b+c x^2\right )+\log (x) \]
[Out]
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Rubi [A] time = 0.0598919, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{1}{2} \log \left (b+c x^2\right )+\log (x) \]
Antiderivative was successfully verified.
[In] Int[(b + 2*c*x^2)/(x*(b + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 10.0293, size = 15, normalized size = 1. \[ \frac{\log{\left (x^{2} \right )}}{2} + \frac{\log{\left (b + c x^{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x**2+b)/x/(c*x**2+b),x)
[Out]
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Mathematica [A] time = 0.0117466, size = 15, normalized size = 1. \[ \frac{1}{2} \log \left (b+c x^2\right )+\log (x) \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*c*x^2)/(x*(b + c*x^2)),x]
[Out]
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Maple [A] time = 0.007, size = 14, normalized size = 0.9 \[ \ln \left ( x \right ) +{\frac{\ln \left ( c{x}^{2}+b \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x^2+b)/x/(c*x^2+b),x)
[Out]
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Maxima [A] time = 1.39064, size = 23, normalized size = 1.53 \[ \frac{1}{2} \, \log \left (c x^{2} + b\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^2 + b)/((c*x^2 + b)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211203, size = 18, normalized size = 1.2 \[ \frac{1}{2} \, \log \left (c x^{2} + b\right ) + \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^2 + b)/((c*x^2 + b)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.61957, size = 12, normalized size = 0.8 \[ \log{\left (x \right )} + \frac{\log{\left (\frac{b}{c} + x^{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x**2+b)/x/(c*x**2+b),x)
[Out]
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GIAC/XCAS [A] time = 0.210649, size = 24, normalized size = 1.6 \[ \frac{1}{2} \,{\rm ln}\left (x^{2}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | c x^{2} + b \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^2 + b)/((c*x^2 + b)*x),x, algorithm="giac")
[Out]