Optimal. Leaf size=113 \[ \frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt [3]{b}}+\frac {\sqrt [3]{x}}{a (a+b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {51, 58, 617, 204, 31} \begin {gather*} \frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt [3]{b}}+\frac {\sqrt [3]{x}}{a (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 51
Rule 58
Rule 204
Rule 617
Rubi steps
\begin {align*} \int \frac {1}{x^{2/3} (a+b x)^2} \, dx &=\frac {\sqrt [3]{x}}{a (a+b x)}+\frac {2 \int \frac {1}{x^{2/3} (a+b x)} \, dx}{3 a}\\ &=\frac {\sqrt [3]{x}}{a (a+b x)}-\frac {\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{x}\right )}{a^{4/3} b^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}\\ &=\frac {\sqrt [3]{x}}{a (a+b x)}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{a^{5/3} \sqrt [3]{b}}\\ &=\frac {\sqrt [3]{x}}{a (a+b x)}-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{5/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 25, normalized size = 0.22 \begin {gather*} \frac {3 \sqrt [3]{x} \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};-\frac {b x}{a}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 144, normalized size = 1.27 \begin {gather*} -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{3 a^{5/3} \sqrt [3]{b}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{5/3} \sqrt [3]{b}}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt [3]{b}}+\frac {\sqrt [3]{x}}{a (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.19, size = 387, normalized size = 3.42 \begin {gather*} \left [\frac {3 \, a^{2} b x^{\frac {1}{3}} + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{\frac {2}{3}} - \left (a^{2} b\right )^{\frac {1}{3}} a + \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x^{\frac {1}{3}}}{b x + a}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{3 \, {\left (a^{3} b^{2} x + a^{4} b\right )}}, \frac {3 \, a^{2} b x^{\frac {1}{3}} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (a^{2} b\right )^{\frac {1}{3}} a - 2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{3 \, {\left (a^{3} b^{2} x + a^{4} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 132, normalized size = 1.17 \begin {gather*} -\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} + \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} + \frac {x^{\frac {1}{3}}}{{\left (b x + a\right )} a} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 120, normalized size = 1.06 \begin {gather*} \frac {x^{\frac {1}{3}}}{\left (b x +a \right ) a}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}+\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 127, normalized size = 1.12 \begin {gather*} \frac {x^{\frac {1}{3}}}{a b x + a^{2}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 134, normalized size = 1.19 \begin {gather*} \frac {2\,\ln \left (\frac {6\,b^{5/3}}{a^{2/3}}+\frac {6\,b^2\,x^{1/3}}{a}\right )}{3\,a^{5/3}\,b^{1/3}}+\frac {x^{1/3}}{a\,\left (a+b\,x\right )}+\frac {\ln \left (\frac {6\,b^2\,x^{1/3}}{a}+\frac {3\,b^{5/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{a^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{3\,a^{5/3}\,b^{1/3}}-\frac {\ln \left (\frac {6\,b^2\,x^{1/3}}{a}-\frac {3\,b^{5/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{a^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{3\,a^{5/3}\,b^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 115.98, size = 590, normalized size = 5.22 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 \sqrt [3]{x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {3}{5 b^{2} x^{\frac {5}{3}}} & \text {for}\: a = 0 \\- \frac {2 \sqrt [3]{-1} a^{\frac {4}{3}} \sqrt [3]{\frac {1}{b}} \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + \sqrt [3]{x} \right )}}{3 a^{3} + 3 a^{2} b x} + \frac {\sqrt [3]{-1} a^{\frac {4}{3}} \sqrt [3]{\frac {1}{b}} \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{\frac {1}{b}} + 4 x^{\frac {2}{3}} \right )}}{3 a^{3} + 3 a^{2} b x} + \frac {2 \sqrt [3]{-1} \sqrt {3} a^{\frac {4}{3}} \sqrt [3]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \left (-1\right )^{\frac {2}{3}} \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{a} \sqrt [3]{\frac {1}{b}}} \right )}}{3 a^{3} + 3 a^{2} b x} - \frac {2 \sqrt [3]{-1} a^{\frac {4}{3}} \sqrt [3]{\frac {1}{b}} \log {\relax (2 )}}{3 a^{3} + 3 a^{2} b x} - \frac {2 \sqrt [3]{-1} \sqrt [3]{a} b x \sqrt [3]{\frac {1}{b}} \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + \sqrt [3]{x} \right )}}{3 a^{3} + 3 a^{2} b x} + \frac {\sqrt [3]{-1} \sqrt [3]{a} b x \sqrt [3]{\frac {1}{b}} \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{\frac {1}{b}} + 4 x^{\frac {2}{3}} \right )}}{3 a^{3} + 3 a^{2} b x} + \frac {2 \sqrt [3]{-1} \sqrt {3} \sqrt [3]{a} b x \sqrt [3]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \left (-1\right )^{\frac {2}{3}} \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{a} \sqrt [3]{\frac {1}{b}}} \right )}}{3 a^{3} + 3 a^{2} b x} - \frac {2 \sqrt [3]{-1} \sqrt [3]{a} b x \sqrt [3]{\frac {1}{b}} \log {\relax (2 )}}{3 a^{3} + 3 a^{2} b x} + \frac {3 a \sqrt [3]{x}}{3 a^{3} + 3 a^{2} b x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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