Optimal. Leaf size=56 \[ -\frac {3 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac {3}{b^2 \left (a \sqrt [3]{x}+b\right )}+\frac {3}{2 b \left (a \sqrt [3]{x}+b\right )^2}+\frac {\log (x)}{b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {263, 266, 44} \[ \frac {3}{b^2 \left (a \sqrt [3]{x}+b\right )}-\frac {3 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac {3}{2 b \left (a \sqrt [3]{x}+b\right )^2}+\frac {\log (x)}{b^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2} \, dx &=\int \frac {1}{\left (b+a \sqrt [3]{x}\right )^3 x} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {1}{x (b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{b^3 x}-\frac {a}{b (b+a x)^3}-\frac {a}{b^2 (b+a x)^2}-\frac {a}{b^3 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{2 b \left (b+a \sqrt [3]{x}\right )^2}+\frac {3}{b^2 \left (b+a \sqrt [3]{x}\right )}-\frac {3 \log \left (b+a \sqrt [3]{x}\right )}{b^3}+\frac {\log (x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 50, normalized size = 0.89 \[ \frac {\frac {3 b \left (2 a \sqrt [3]{x}+3 b\right )}{\left (a \sqrt [3]{x}+b\right )^2}-6 \log \left (a \sqrt [3]{x}+b\right )+2 \log (x)}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 129, normalized size = 2.30 \[ \frac {3 \, {\left (3 \, b^{6} - 2 \, {\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 2 \, {\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (x^{\frac {1}{3}}\right ) + {\left (2 \, a^{5} b x + 5 \, a^{2} b^{4}\right )} x^{\frac {2}{3}} - {\left (a^{4} b^{2} x + 4 \, a b^{5}\right )} x^{\frac {1}{3}}\right )}}{2 \, {\left (a^{6} b^{3} x^{2} + 2 \, a^{3} b^{6} x + b^{9}\right )}} \]
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 \[ -\frac {3 \, \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{3}} + \frac {\log \left ({\left | x \right |}\right )}{b^{3}} + \frac {3 \, {\left (2 \, a b x^{\frac {1}{3}} + 3 \, b^{2}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} b^{3}} \]
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 \[ \frac {3}{2 \left (a \,x^{\frac {1}{3}}+b \right )^{2} b}+\frac {3}{\left (a \,x^{\frac {1}{3}}+b \right ) b^{2}}+\frac {\ln \relax (x )}{b^{3}}-\frac {3 \ln \left (a \,x^{\frac {1}{3}}+b \right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 46, normalized size = 0.82 \[ -\frac {3 \, \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{3}} - \frac {6 \, a}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{3}} + \frac {3 \, a^{2}}{2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} b^{3}} \]
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 \[ \frac {\frac {9}{2\,b}+\frac {3\,a\,x^{1/3}}{b^2}}{b^2+a^2\,x^{2/3}+2\,a\,b\,x^{1/3}}-\frac {6\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.63, size = 406, normalized size = 7.25 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\relax (x )}}{b^{3}} & \text {for}\: a = 0 \\- \frac {1}{a^{3} x} & \text {for}\: b = 0 \\\frac {2 a^{2} x^{\frac {7}{3}} \log {\relax (x )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {6 a^{2} x^{\frac {7}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {4 a b x^{2} \log {\relax (x )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {12 a b x^{2} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {6 a b x^{2}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {2 b^{2} x^{\frac {5}{3}} \log {\relax (x )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} - \frac {6 b^{2} x^{\frac {5}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} + \frac {9 b^{2} x^{\frac {5}{3}}}{2 a^{2} b^{3} x^{\frac {7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac {5}{3}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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