3.2399 \(\int \frac{a+\frac{b}{\sqrt [3]{x}}}{x} \, dx\)

Optimal. Leaf size=13 \[ a \log (x)-\frac{3 b}{\sqrt [3]{x}} \]

[Out]

(-3*b)/x^(1/3) + a*Log[x]

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Rubi [A]  time = 0.0053345, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ a \log (x)-\frac{3 b}{\sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))/x,x]

[Out]

(-3*b)/x^(1/3) + a*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{a+\frac{b}{\sqrt [3]{x}}}{x} \, dx &=\int \left (\frac{b}{x^{4/3}}+\frac{a}{x}\right ) \, dx\\ &=-\frac{3 b}{\sqrt [3]{x}}+a \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0068999, size = 13, normalized size = 1. \[ a \log (x)-\frac{3 b}{\sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))/x,x]

[Out]

(-3*b)/x^(1/3) + a*Log[x]

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Maple [A]  time = 0.005, size = 12, normalized size = 0.9 \begin{align*} -3\,{\frac{b}{\sqrt [3]{x}}}+a\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^(1/3))/x,x)

[Out]

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

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Maxima [A]  time = 0.963274, size = 15, normalized size = 1.15 \begin{align*} a \log \left (x\right ) - \frac{3 \, b}{x^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) - 3*b/x^(1/3)

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Fricas [A]  time = 1.52474, size = 49, normalized size = 3.77 \begin{align*} \frac{3 \,{\left (a x \log \left (x^{\frac{1}{3}}\right ) - b x^{\frac{2}{3}}\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(a*x*log(x^(1/3)) - b*x^(2/3))/x

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Sympy [A]  time = 0.381912, size = 12, normalized size = 0.92 \begin{align*} a \log{\left (x \right )} - \frac{3 b}{\sqrt [3]{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**(1/3))/x,x)

[Out]

a*log(x) - 3*b/x**(1/3)

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Giac [A]  time = 1.17243, size = 16, normalized size = 1.23 \begin{align*} a \log \left ({\left | x \right |}\right ) - \frac{3 \, b}{x^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(abs(x)) - 3*b/x^(1/3)