3.24.99 \(\int \frac {a+\frac {b}{\sqrt [3]{x}}}{x} \, dx\) [2399]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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} \begin {gather*} a \log (x)-\frac {3 b}{\sqrt [3]{x}} \end {gather*}

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*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} -\frac {3 b}{\sqrt [3]{x}}+a \log (x) \end {gather*}

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 = 0.05, size = 12, normalized size = 0.92

method	result	size
derivativedivides	\(-\frac {3 b}{x^{\frac {1}{3}}}+a \ln \left (x \right )\)	\(12\)
default	\(-\frac {3 b}{x^{\frac {1}{3}}}+a \ln \left (x \right )\)	\(12\)
trager	\(-\frac {3 b}{x^{\frac {1}{3}}}+a \ln \left (x \right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 11, normalized size = 0.85 \begin {gather*} a \log \left (x\right ) - \frac {3 \, b}{x^{\frac {1}{3}}} \end {gather*}

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 = 0.37, size = 19, normalized size = 1.46 \begin {gather*} \frac {3 \, {\left (a x \log \left (x^{\frac {1}{3}}\right ) - b x^{\frac {2}{3}}\right )}}{x} \end {gather*}

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 = 0.15, size = 12, normalized size = 0.92 \begin {gather*} a \log {\left (x \right )} - \frac {3 b}{\sqrt [3]{x}} \end {gather*}

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 = 0.54, size = 12, normalized size = 0.92 \begin {gather*} a \log \left ({\left | x \right |}\right ) - \frac {3 \, b}{x^{\frac {1}{3}}} \end {gather*}

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)

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Mupad [B]
time = 0.04, size = 14, normalized size = 1.08 \begin {gather*} 3\,a\,\ln \left (x^{1/3}\right )-\frac {3\,b}{x^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^(1/3))/x,x)

[Out]

3*a*log(x^(1/3)) - (3*b)/x^(1/3)

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