Optimal. Leaf size=56 \[ -\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{a^3}+\frac {\log (x)}{a^3}+\frac {3}{a^2 \left (a+b \sqrt [3]{x}\right )}+\frac {3}{2 a \left (a+b \sqrt [3]{x}\right )^2} \]
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Rubi [A] time = 0.03, antiderivative size = 56, 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} \[ \frac {3}{a^2 \left (a+b \sqrt [3]{x}\right )}-\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{a^3}+\frac {\log (x)}{a^3}+\frac {3}{2 a \left (a+b \sqrt [3]{x}\right )^2} \]
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 )^3 x} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{a^3 x}-\frac {b}{a (a+b x)^3}-\frac {b}{a^2 (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{2 a \left (a+b \sqrt [3]{x}\right )^2}+\frac {3}{a^2 \left (a+b \sqrt [3]{x}\right )}-\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{a^3}+\frac {\log (x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 50, normalized size = 0.89 \[ \frac {\frac {3 a \left (3 a+2 b \sqrt [3]{x}\right )}{\left (a+b \sqrt [3]{x}\right )^2}-6 \log \left (a+b \sqrt [3]{x}\right )+2 \log (x)}{2 a^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 129, normalized size = 2.30 \[ \frac {3 \, {\left (3 \, a^{6} - 2 \, {\left (b^{6} x^{2} + 2 \, a^{3} b^{3} x + a^{6}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + 2 \, {\left (b^{6} x^{2} + 2 \, a^{3} b^{3} x + a^{6}\right )} \log \left (x^{\frac {1}{3}}\right ) + {\left (2 \, a b^{5} x + 5 \, a^{4} b^{2}\right )} x^{\frac {2}{3}} - {\left (a^{2} b^{4} x + 4 \, a^{5} b\right )} x^{\frac {1}{3}}\right )}}{2 \, {\left (a^{3} b^{6} x^{2} + 2 \, a^{6} b^{3} x + a^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 49, normalized size = 0.88 \[ -\frac {3 \, \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} + \frac {3 \, {\left (2 \, a b x^{\frac {1}{3}} + 3 \, a^{2}\right )}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 49, normalized size = 0.88 \[ \frac {3}{2 \left (b \,x^{\frac {1}{3}}+a \right )^{2} a}+\frac {3}{\left (b \,x^{\frac {1}{3}}+a \right ) a^{2}}+\frac {\ln \relax (x )}{a^{3}}-\frac {3 \ln \left (b \,x^{\frac {1}{3}}+a \right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 57, normalized size = 1.02 \[ \frac {3 \, {\left (2 \, b x^{\frac {1}{3}} + 3 \, a\right )}}{2 \, {\left (a^{2} b^{2} x^{\frac {2}{3}} + 2 \, a^{3} b x^{\frac {1}{3}} + a^{4}\right )}} - \frac {3 \, \log \left (b x^{\frac {1}{3}} + a\right )}{a^{3}} + \frac {\log \relax (x)}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 54, normalized size = 0.96 \[ \frac {\frac {9}{2\,a}+\frac {3\,b\,x^{1/3}}{a^2}}{a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}-\frac {6\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.91, size = 386, normalized size = 6.89 \[ \begin {cases} \frac {\tilde {\infty }}{x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\relax (x )}}{a^{3}} & \text {for}\: b = 0 \\- \frac {1}{b^{3} x} & \text {for}\: a = 0 \\\frac {2 a^{2} x^{\frac {2}{3}} \log {\relax (x )}}{2 a^{5} x^{\frac {2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {4}{3}}} - \frac {6 a^{2} x^{\frac {2}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{5} x^{\frac {2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {4}{3}}} + \frac {9 a^{2} x^{\frac {2}{3}}}{2 a^{5} x^{\frac {2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {4}{3}}} + \frac {4 a b x \log {\relax (x )}}{2 a^{5} x^{\frac {2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {4}{3}}} - \frac {12 a b x \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{5} x^{\frac {2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {4}{3}}} + \frac {6 a b x}{2 a^{5} x^{\frac {2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {4}{3}}} + \frac {2 b^{2} x^{\frac {4}{3}} \log {\relax (x )}}{2 a^{5} x^{\frac {2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {4}{3}}} - \frac {6 b^{2} x^{\frac {4}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{5} x^{\frac {2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac {4}{3}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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