3.889 \(\int \frac{b+2 c x^3}{x \left (b+c x^3\right )} \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{3} \log \left (b+c x^3\right )+\log (x) \]

[Out]

Log[x] + Log[b + c*x^3]/3

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Rubi [A]  time = 0.0592244, 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 \[ \frac{1}{3} \log \left (b+c x^3\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x^3)/(x*(b + c*x^3)),x]

[Out]

Log[x] + Log[b + c*x^3]/3

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Rubi in Sympy [A]  time = 9.38372, size = 15, normalized size = 1. \[ \frac{\log{\left (x^{3} \right )}}{3} + \frac{\log{\left (b + c x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x**3+b)/x/(c*x**3+b),x)

[Out]

log(x**3)/3 + log(b + c*x**3)/3

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Mathematica [A]  time = 0.011386, size = 15, normalized size = 1. \[ \frac{1}{3} \log \left (b+c x^3\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(b + 2*c*x^3)/(x*(b + c*x^3)),x]

[Out]

Log[x] + Log[b + c*x^3]/3

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Maple [A]  time = 0.007, size = 14, normalized size = 0.9 \[ \ln \left ( x \right ) +{\frac{\ln \left ( c{x}^{3}+b \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x^3+b)/x/(c*x^3+b),x)

[Out]

ln(x)+1/3*ln(c*x^3+b)

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Maxima [A]  time = 1.38991, size = 23, normalized size = 1.53 \[ \frac{1}{3} \, \log \left (c x^{3} + b\right ) + \frac{1}{3} \, \log \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^3 + b)/((c*x^3 + b)*x),x, algorithm="maxima")

[Out]

1/3*log(c*x^3 + b) + 1/3*log(x^3)

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Fricas [A]  time = 0.207814, size = 18, normalized size = 1.2 \[ \frac{1}{3} \, \log \left (c x^{3} + b\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^3 + b)/((c*x^3 + b)*x),x, algorithm="fricas")

[Out]

1/3*log(c*x^3 + b) + log(x)

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Sympy [A]  time = 0.652618, size = 12, normalized size = 0.8 \[ \log{\left (x \right )} + \frac{\log{\left (\frac{b}{c} + x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x**3+b)/x/(c*x**3+b),x)

[Out]

log(x) + log(b/c + x**3)/3

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GIAC/XCAS [A]  time = 0.211204, size = 20, normalized size = 1.33 \[ \frac{1}{3} \,{\rm ln}\left ({\left | c x^{3} + b \right |}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^3 + b)/((c*x^3 + b)*x),x, algorithm="giac")

[Out]

1/3*ln(abs(c*x^3 + b)) + ln(abs(x))