3.122 \(\int \frac{x \left (b+2 c x^2\right )}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{2} \log \left (b x^2+c x^4\right ) \]

[Out]

Log[b*x^2 + c*x^4]/2

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Rubi [A]  time = 0.0089944, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{1}{2} \log \left (b x^2+c x^4\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x*(b + 2*c*x^2))/(b*x^2 + c*x^4),x]

[Out]

Log[b*x^2 + c*x^4]/2

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Rubi in Sympy [A]  time = 11.5813, size = 15, normalized size = 0.94 \[ \frac{\log{\left (x^{2} \right )}}{2} + \frac{\log{\left (b + c x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(2*c*x**2+b)/(c*x**4+b*x**2),x)

[Out]

log(x**2)/2 + log(b + c*x**2)/2

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Mathematica [A]  time = 0.0100363, size = 15, normalized size = 0.94 \[ \frac{1}{2} \log \left (b+c x^2\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(b + 2*c*x^2))/(b*x^2 + c*x^4),x]

[Out]

Log[x] + Log[b + c*x^2]/2

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Maple [A]  time = 0.007, size = 14, normalized size = 0.9 \[{\frac{\ln \left ( c{x}^{2}+b \right ) }{2}}+\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(2*c*x^2+b)/(c*x^4+b*x^2),x)

[Out]

1/2*ln(c*x^2+b)+ln(x)

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Maxima [A]  time = 0.750682, size = 23, normalized size = 1.44 \[ \frac{1}{2} \, \log \left (c x^{2} + b\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 + b)*x/(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

1/2*log(c*x^2 + b) + 1/2*log(x^2)

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Fricas [A]  time = 0.282258, size = 18, normalized size = 1.12 \[ \frac{1}{2} \, \log \left (c x^{2} + b\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 + b)*x/(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

1/2*log(c*x^2 + b) + log(x)

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Sympy [A]  time = 1.23381, size = 12, normalized size = 0.75 \[ \log{\left (x \right )} + \frac{\log{\left (\frac{b}{c} + x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(2*c*x**2+b)/(c*x**4+b*x**2),x)

[Out]

log(x) + log(b/c + x**2)/2

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GIAC/XCAS [A]  time = 0.263608, size = 20, normalized size = 1.25 \[ \frac{1}{2} \,{\rm ln}\left ({\left | c x^{2} + b \right |}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 + b)*x/(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

1/2*ln(abs(c*x^2 + b)) + ln(abs(x))