3.1969 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right ) x} \, dx\)

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

[Out]

Log[b + a*x^3]/(3*a)

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Rubi [A]  time = 0.0200524, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\log \left (a x^3+b\right )}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)*x),x]

[Out]

Log[b + a*x^3]/(3*a)

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Rubi in Sympy [A]  time = 3.6875, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a x^{3} + b \right )}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*x**3 + b)/(3*a)

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Mathematica [A]  time = 0.00526372, size = 15, normalized size = 1. \[ \frac{\log \left (a x^3+b\right )}{3 a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^3)*x),x]

[Out]

Log[b + a*x^3]/(3*a)

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Maple [A]  time = 0.003, size = 14, normalized size = 0.9 \[{\frac{\ln \left ( a{x}^{3}+b \right ) }{3\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)/x,x)

[Out]

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

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Maxima [A]  time = 1.4243, size = 18, normalized size = 1.2 \[ \frac{\log \left (a x^{3} + b\right )}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log(a*x^3 + b)/a

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Fricas [A]  time = 0.22054, size = 18, normalized size = 1.2 \[ \frac{\log \left (a x^{3} + b\right )}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log(a*x^3 + b)/a

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Sympy [A]  time = 0.322607, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a x^{3} + b \right )}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)/x,x)

[Out]

log(a*x**3 + b)/(3*a)

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GIAC/XCAS [A]  time = 0.232385, size = 19, normalized size = 1.27 \[ \frac{{\rm ln}\left ({\left | a x^{3} + b \right |}\right )}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*ln(abs(a*x^3 + b))/a