3.2722 \(\int \frac{1}{x \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^3\right )}{3 a} \]

[Out]

Log[x]/a - Log[a + b*x^3]/(3*a)

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Rubi [A]  time = 0.0378885, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^3\right )}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^3)),x]

[Out]

Log[x]/a - Log[a + b*x^3]/(3*a)

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Rubi in Sympy [A]  time = 5.43752, size = 19, normalized size = 0.86 \[ \frac{\log{\left (x^{3} \right )}}{3 a} - \frac{\log{\left (a + b x^{3} \right )}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**3+a),x)

[Out]

log(x**3)/(3*a) - log(a + b*x**3)/(3*a)

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Mathematica [A]  time = 0.00889521, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^3\right )}{3 a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^3)),x]

[Out]

Log[x]/a - Log[a + b*x^3]/(3*a)

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Maple [A]  time = 0.001, size = 21, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{3}+a \right ) }{3\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^3+a),x)

[Out]

ln(x)/a-1/3*ln(b*x^3+a)/a

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Maxima [A]  time = 1.34389, size = 31, normalized size = 1.41 \[ -\frac{\log \left (b x^{3} + a\right )}{3 \, a} + \frac{\log \left (x^{3}\right )}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*x),x, algorithm="maxima")

[Out]

-1/3*log(b*x^3 + a)/a + 1/3*log(x^3)/a

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Fricas [A]  time = 0.216405, size = 24, normalized size = 1.09 \[ -\frac{\log \left (b x^{3} + a\right ) - 3 \, \log \left (x\right )}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*x),x, algorithm="fricas")

[Out]

-1/3*(log(b*x^3 + a) - 3*log(x))/a

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Sympy [A]  time = 0.572219, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{3} \right )}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**3+a),x)

[Out]

log(x)/a - log(a/b + x**3)/(3*a)

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GIAC/XCAS [A]  time = 0.217243, size = 30, normalized size = 1.36 \[ -\frac{{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*x),x, algorithm="giac")

[Out]

-1/3*ln(abs(b*x^3 + a))/a + ln(abs(x))/a