Optimal. Leaf size=22 \[ -\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{a}+\frac {\log (x)}{a} \]
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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.
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Rule 29
Rule 31
Rule 36
Rule 272
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.
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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.
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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.
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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.
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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.
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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.
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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.
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