3.24.60 \(\int \frac {1}{(a+b \sqrt [3]{x}) x} \, dx\) [2360]

Optimal. Leaf size=22 \[ -\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{a}+\frac {\log (x)}{a} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {272, 36, 29, 31} \begin {gather*} \frac {\log (x)}{a}-\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))*x),x]

[Out]

(-3*Log[a + b*x^(1/3)])/a + Log[x]/a

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]

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 272

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]]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 30, normalized size = 1.36 \begin {gather*} -\frac {3 \log \left (a^2+a b \sqrt [3]{x}\right )}{a}+\frac {3 \log \left (\sqrt [3]{x}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^(1/3))*x),x]

[Out]

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

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Maple [A]
time = 0.20, size = 21, normalized size = 0.95

method	result	size
derivativedivides	\(-\frac {3 \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a}+\frac {\ln \left (x \right )}{a}\)	\(21\)
default	\(-\frac {3 \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a}+\frac {\ln \left (x \right )}{a}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 20, normalized size = 0.91 \begin {gather*} -\frac {3 \, \log \left (b x^{\frac {1}{3}} + a\right )}{a} + \frac {\log \left (x\right )}{a} \end {gather*}

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(b*x^(1/3) + a)/a + log(x)/a

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Fricas [A]
time = 0.38, size = 20, normalized size = 0.91 \begin {gather*} -\frac {3 \, {\left (\log \left (b x^{\frac {1}{3}} + a\right ) - \log \left (x^{\frac {1}{3}}\right )\right )}}{a} \end {gather*}

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(b*x^(1/3) + a) - log(x^(1/3)))/a

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
time = 0.16, size = 37, normalized size = 1.68 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{\sqrt [3]{x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\- \frac {3}{b \sqrt [3]{x}} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a} - \frac {3 \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{a} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(1/3), Eq(a, 0) & Eq(b, 0)), (log(x)/a, Eq(b, 0)), (-3/(b*x**(1/3)), Eq(a, 0)), (log(x)/a -
3*log(a/b + x**(1/3))/a, True))

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Giac [A]
time = 1.13, size = 22, normalized size = 1.00 \begin {gather*} -\frac {3 \, \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a} + \frac {\log \left ({\left | x \right |}\right )}{a} \end {gather*}

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(b*x^(1/3) + a))/a + log(abs(x))/a

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Mupad [B]
time = 1.21, size = 17, normalized size = 0.77 \begin {gather*} -\frac {6\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^(1/3))),x)

[Out]

-(6*atanh((2*b*x^(1/3))/a + 1))/a

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