3.2417 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right ) x} \, dx\)

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

[Out]

(3*Log[b + a*x^(1/3)])/a

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Rubi [A]  time = 0.0231028, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^(1/3))*x),x]

[Out]

(3*Log[b + a*x^(1/3)])/a

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Rubi in Sympy [A]  time = 4.04181, size = 12, normalized size = 0.8 \[ \frac{3 \log{\left (a \sqrt [3]{x} + b \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(a*x**(1/3) + b)/a

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Mathematica [A]  time = 0.00711098, size = 15, normalized size = 1. \[ \frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

(3*Log[b + a*x^(1/3)])/a

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Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \[ 3\,{\frac{\ln \left ( b+a\sqrt [3]{x} \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))/x,x)

[Out]

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

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Maxima [A]  time = 1.41762, size = 27, normalized size = 1.8 \[ \frac{3 \, \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a} + \frac{\log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(a + b/x^(1/3))/a + log(x)/a

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Fricas [A]  time = 0.228615, size = 18, normalized size = 1.2 \[ \frac{3 \, \log \left (a x^{\frac{1}{3}} + b\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(a*x^(1/3) + b)/a

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Sympy [A]  time = 2.35549, size = 20, normalized size = 1.33 \[ \begin{cases} \frac{3 \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a} & \text{for}\: a \neq 0 \\\frac{3 \sqrt [3]{x}}{b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**(1/3))/x,x)

[Out]

Piecewise((3*log(x**(1/3) + b/a)/a, Ne(a, 0)), (3*x**(1/3)/b, True))

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GIAC/XCAS [A]  time = 0.213621, size = 19, normalized size = 1.27 \[ \frac{3 \,{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*ln(abs(a*x^(1/3) + b))/a