3.2357 \(\int \frac{1}{\left (2+b \sqrt [3]{x}\right ) x} \, dx\)

Optimal. Leaf size=21 \[ \frac{\log (x)}{2}-\frac{3}{2} \log \left (b \sqrt [3]{x}+2\right ) \]

[Out]

(-3*Log[2 + b*x^(1/3)])/2 + Log[x]/2

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Rubi [A]  time = 0.0276904, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\log (x)}{2}-\frac{3}{2} \log \left (b \sqrt [3]{x}+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((2 + b*x^(1/3))*x),x]

[Out]

(-3*Log[2 + b*x^(1/3)])/2 + Log[x]/2

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Rubi in Sympy [A]  time = 4.59721, size = 22, normalized size = 1.05 \[ \frac{3 \log{\left (\sqrt [3]{x} \right )}}{2} - \frac{3 \log{\left (b \sqrt [3]{x} + 2 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2+b*x**(1/3))/x,x)

[Out]

3*log(x**(1/3))/2 - 3*log(b*x**(1/3) + 2)/2

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Mathematica [A]  time = 0.00922767, size = 25, normalized size = 1.19 \[ \frac{3}{2} \log \left (\sqrt [3]{x}\right )-\frac{3}{2} \log \left (b \sqrt [3]{x}+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((2 + b*x^(1/3))*x),x]

[Out]

(-3*Log[2 + b*x^(1/3)])/2 + (3*Log[x^(1/3)])/2

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Maple [A]  time = 0.004, size = 16, normalized size = 0.8 \[ -{\frac{3}{2}\ln \left ( 2+b\sqrt [3]{x} \right ) }+{\frac{\ln \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2+b*x^(1/3))/x,x)

[Out]

-3/2*ln(2+b*x^(1/3))+1/2*ln(x)

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Maxima [A]  time = 1.4382, size = 20, normalized size = 0.95 \[ -\frac{3}{2} \, \log \left (b x^{\frac{1}{3}} + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^(1/3) + 2)*x),x, algorithm="maxima")

[Out]

-3/2*log(b*x^(1/3) + 2) + 1/2*log(x)

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Fricas [A]  time = 0.219521, size = 23, normalized size = 1.1 \[ -\frac{3}{2} \, \log \left (b x^{\frac{1}{3}} + 2\right ) + \frac{3}{2} \, \log \left (x^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^(1/3) + 2)*x),x, algorithm="fricas")

[Out]

-3/2*log(b*x^(1/3) + 2) + 3/2*log(x^(1/3))

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Sympy [A]  time = 2.265, size = 22, normalized size = 1.05 \[ \begin{cases} \frac{\log{\left (x \right )}}{2} - \frac{3 \log{\left (\sqrt [3]{x} + \frac{2}{b} \right )}}{2} & \text{for}\: b \neq 0 \\\frac{\log{\left (x \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2+b*x**(1/3))/x,x)

[Out]

Piecewise((log(x)/2 - 3*log(x**(1/3) + 2/b)/2, Ne(b, 0)), (log(x)/2, True))

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GIAC/XCAS [A]  time = 0.220211, size = 23, normalized size = 1.1 \[ -\frac{3}{2} \,{\rm ln}\left ({\left | b x^{\frac{1}{3}} + 2 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^(1/3) + 2)*x),x, algorithm="giac")

[Out]

-3/2*ln(abs(b*x^(1/3) + 2)) + 1/2*ln(abs(x))