3.2364 \(\int \frac{1}{(2+b \sqrt [3]{x}) x} \, dx\)

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 = 0.0077708, 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, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{2}-\frac{3}{2} \log \left (b \sqrt [3]{x}+2\right ) \]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (2+b \sqrt [3]{x}\right ) x} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x (2+b x)} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sqrt [3]{x}\right )-\frac{1}{2} (3 b) \operatorname{Subst}\left (\int \frac{1}{2+b x} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3}{2} \log \left (2+b \sqrt [3]{x}\right )+\frac{\log (x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.0041928, size = 21, normalized size = 1. \[ \frac{\log (x)}{2}-\frac{3}{2} \log \left (b \sqrt [3]{x}+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((2 + b*x^(1/3))*x),x]

[Out]

(-3*Log[2 + b*x^(1/3)])/2 + Log[x]/2

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \begin{align*} -{\frac{3}{2}\ln \left ( 2+b\sqrt [3]{x} \right ) }+{\frac{\ln \left ( x \right ) }{2}} \end{align*}

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 = 0.964906, size = 20, normalized size = 0.95 \begin{align*} -\frac{3}{2} \, \log \left (b x^{\frac{1}{3}} + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*log(b*x^(1/3) + 2) + 1/2*log(x)

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Fricas [A]  time = 1.50683, size = 59, normalized size = 2.81 \begin{align*} -\frac{3}{2} \, \log \left (b x^{\frac{1}{3}} + 2\right ) + \frac{3}{2} \, \log \left (x^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+b*x^(1/3))/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 = 0.572891, size = 22, normalized size = 1.05 \begin{align*} \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} \end{align*}

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 [A]  time = 1.15046, size = 23, normalized size = 1.1 \begin{align*} -\frac{3}{2} \, \log \left ({\left | b x^{\frac{1}{3}} + 2 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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