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

Optimal. Leaf size=22 \[ \frac{\log (x)}{b}-\frac{\log \left (b+c x^2\right )}{2 b} \]

[Out]

Log[x]/b - Log[b + c*x^2]/(2*b)

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Rubi [A]  time = 0.038038, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\log (x)}{b}-\frac{\log \left (b+c x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x/(b*x^2 + c*x^4),x]

[Out]

Log[x]/b - Log[b + c*x^2]/(2*b)

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Rubi in Sympy [A]  time = 8.41414, size = 19, normalized size = 0.86 \[ \frac{\log{\left (x^{2} \right )}}{2 b} - \frac{\log{\left (b + c x^{2} \right )}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2)/(2*b) - log(b + c*x**2)/(2*b)

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Mathematica [A]  time = 0.00836916, size = 22, normalized size = 1. \[ \frac{\log (x)}{b}-\frac{\log \left (b+c x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(b*x^2 + c*x^4),x]

[Out]

Log[x]/b - Log[b + c*x^2]/(2*b)

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Maple [A]  time = 0.007, size = 21, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( c{x}^{2}+b \right ) }{2\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c*x^4+b*x^2),x)

[Out]

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

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Maxima [A]  time = 0.680484, size = 31, normalized size = 1.41 \[ -\frac{\log \left (c x^{2} + b\right )}{2 \, b} + \frac{\log \left (x^{2}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(c*x^2 + b)/b + 1/2*log(x^2)/b

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Fricas [A]  time = 0.255674, size = 24, normalized size = 1.09 \[ -\frac{\log \left (c x^{2} + b\right ) - 2 \, \log \left (x\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(log(c*x^2 + b) - 2*log(x))/b

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Sympy [A]  time = 0.505183, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x \right )}}{b} - \frac{\log{\left (\frac{b}{c} + x^{2} \right )}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c*x**4+b*x**2),x)

[Out]

log(x)/b - log(b/c + x**2)/(2*b)

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GIAC/XCAS [A]  time = 0.271004, size = 30, normalized size = 1.36 \[ -\frac{{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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