Optimal. Leaf size=33 \[ \frac {3 b}{a^2 \left (a \sqrt [3]{x}+b\right )}+\frac {3 \log \left (a \sqrt [3]{x}+b\right )}{a^2} \]
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Rubi [A] time = 0.02, antiderivative size = 33, 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, 43} \[ \frac {3 b}{a^2 \left (a \sqrt [3]{x}+b\right )}+\frac {3 \log \left (a \sqrt [3]{x}+b\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x} \, dx &=\int \frac {1}{\left (b+a \sqrt [3]{x}\right )^2 \sqrt [3]{x}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {x}{(b+a x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (-\frac {b}{a (b+a x)^2}+\frac {1}{a (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 b}{a^2 \left (b+a \sqrt [3]{x}\right )}+\frac {3 \log \left (b+a \sqrt [3]{x}\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 1.00 \[ \frac {-\frac {3 a}{a+\frac {b}{\sqrt [3]{x}}}+3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )+\log (x)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 56, normalized size = 1.70 \[ \frac {3 \, {\left (a^{2} b x^{\frac {2}{3}} - a b^{2} x^{\frac {1}{3}} + b^{3} + {\left (a^{3} x + b^{3}\right )} \log \left (a x^{\frac {1}{3}} + b\right )\right )}}{a^{5} x + a^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 30, normalized size = 0.91 \[ \frac {3 \, \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{2}} + \frac {3 \, b}{{\left (a x^{\frac {1}{3}} + b\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.91 \[ \frac {3 b}{\left (a \,x^{\frac {1}{3}}+b \right ) a^{2}}+\frac {3 \ln \left (a \,x^{\frac {1}{3}}+b \right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 34, normalized size = 1.03 \[ -\frac {3}{a^{2} + \frac {a b}{x^{\frac {1}{3}}}} + \frac {3 \, \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{2}} + \frac {\log \relax (x)}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 29, normalized size = 0.88 \[ \frac {3\,\ln \left (b+a\,x^{1/3}\right )}{a^2}+\frac {3\,b}{a^2\,\left (b+a\,x^{1/3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.08, size = 99, normalized size = 3.00 \[ \begin {cases} \frac {3 a x \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{3} x + a^{2} b x^{\frac {2}{3}}} + \frac {3 b x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{3} x + a^{2} b x^{\frac {2}{3}}} + \frac {3 b x^{\frac {2}{3}}}{a^{3} x + a^{2} b x^{\frac {2}{3}}} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {2}{3}}}{2 b^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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