3.2424 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}}) x} \, dx\)

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 = 0.0059596, 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, Rules used = {263, 260} \[ \frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))*x),x]

[Out]

(3*Log[b + a*x^(1/3)])/a

Rule 263

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

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.0030245, 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 = 0.002, size = 14, normalized size = 0.9 \begin{align*} 3\,{\frac{\ln \left ( b+a\sqrt [3]{x} \right ) }{a}} \end{align*}

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 = 0.959822, size = 27, normalized size = 1.8 \begin{align*} \frac{3 \, \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a} + \frac{\log \left (x\right )}{a} \end{align*}

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 = 1.48365, size = 32, normalized size = 2.13 \begin{align*} \frac{3 \, \log \left (a x^{\frac{1}{3}} + b\right )}{a} \end{align*}

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 = 0.864197, size = 20, normalized size = 1.33 \begin{align*} \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} \end{align*}

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 [A]  time = 1.18845, size = 19, normalized size = 1.27 \begin{align*} \frac{3 \, \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(abs(a*x^(1/3) + b))/a