Optimal. Leaf size=38 \[ -\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{a^2}+\frac {\log (x)}{a^2}+\frac {3}{a \left (a+b \sqrt [3]{x}\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 44} \[ -\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{a^2}+\frac {\log (x)}{a^2}+\frac {3}{a \left (a+b \sqrt [3]{x}\right )} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{a \left (a+b \sqrt [3]{x}\right )}-\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{a^2}+\frac {\log (x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 0.87 \[ \frac {\frac {3 a}{a+b \sqrt [3]{x}}-3 \log \left (a+b \sqrt [3]{x}\right )+\log (x)}{a^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 70, normalized size = 1.84 \[ \frac {3 \, {\left (a b^{2} x^{\frac {2}{3}} - a^{2} b x^{\frac {1}{3}} + a^{3} - {\left (b^{3} x + a^{3}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + {\left (b^{3} x + a^{3}\right )} \log \left (x^{\frac {1}{3}}\right )\right )}}{a^{2} b^{3} x + a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 36, normalized size = 0.95 \[ -\frac {3 \, \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a^{2}} + \frac {3}{{\left (b x^{\frac {1}{3}} + a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.92 \[ \frac {3}{\left (b \,x^{\frac {1}{3}}+a \right ) a}+\frac {\ln \relax (x )}{a^{2}}-\frac {3 \ln \left (b \,x^{\frac {1}{3}}+a \right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 34, normalized size = 0.89 \[ \frac {3}{a b x^{\frac {1}{3}} + a^{2}} - \frac {3 \, \log \left (b x^{\frac {1}{3}} + a\right )}{a^{2}} + \frac {\log \relax (x)}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 32, normalized size = 0.84 \[ \frac {3}{a\,\left (a+b\,x^{1/3}\right )}-\frac {6\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.38, size = 160, normalized size = 4.21 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {2}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\relax (x )}}{a^{2}} & \text {for}\: b = 0 \\- \frac {3}{2 b^{2} x^{\frac {2}{3}}} & \text {for}\: a = 0 \\\frac {a x^{\frac {2}{3}} \log {\relax (x )}}{a^{3} x^{\frac {2}{3}} + a^{2} b x} - \frac {3 a x^{\frac {2}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{a^{3} x^{\frac {2}{3}} + a^{2} b x} + \frac {3 a x^{\frac {2}{3}}}{a^{3} x^{\frac {2}{3}} + a^{2} b x} + \frac {b x \log {\relax (x )}}{a^{3} x^{\frac {2}{3}} + a^{2} b x} - \frac {3 b x \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{a^{3} x^{\frac {2}{3}} + a^{2} b x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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