Integrand size = 13, antiderivative size = 113 \[ \int \frac {1}{x^{2/3} (a+b x)^2} \, dx=\frac {\sqrt [3]{x}}{a (a+b x)}-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}} \]
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Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {44, 60, 631, 210, 31} \[ \int \frac {1}{x^{2/3} (a+b x)^2} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{5/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{3 a^{5/3} \sqrt [3]{b}}+\frac {\sqrt [3]{x}}{a (a+b x)} \]
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Rule 31
Rule 44
Rule 60
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = \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 {\text {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 {\text {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 \text {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*}
Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^{2/3} (a+b x)^2} \, dx=\frac {\frac {3 a^{2/3} \sqrt [3]{x}}{a+b x}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{\sqrt [3]{b}}}{3 a^{5/3}} \]
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Time = 0.10 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {x^{\frac {1}{3}}}{a \left (b x +a \right )}+\frac {\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{a}\) | \(117\) |
default | \(\frac {x^{\frac {1}{3}}}{a \left (b x +a \right )}+\frac {\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{a}\) | \(117\) |
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Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (82) = 164\).
Time = 0.24 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.42 \[ \int \frac {1}{x^{2/3} (a+b x)^2} \, dx=\left [\frac {3 \, a^{2} b x^{\frac {1}{3}} + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{\frac {2}{3}} - \left (a^{2} b\right )^{\frac {1}{3}} a + \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x^{\frac {1}{3}}}{b x + a}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{3 \, {\left (a^{3} b^{2} x + a^{4} b\right )}}, \frac {3 \, a^{2} b x^{\frac {1}{3}} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (a^{2} b\right )^{\frac {1}{3}} a - 2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{3 \, {\left (a^{3} b^{2} x + a^{4} b\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (107) = 214\).
Time = 151.97 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.84 \[ \int \frac {1}{x^{2/3} (a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 \sqrt [3]{x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {3}{5 b^{2} x^{\frac {5}{3}}} & \text {for}\: a = 0 \\\frac {3 a \sqrt [3]{x}}{3 a^{3} + 3 a^{2} b x} - \frac {2 a \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{3 a^{3} + 3 a^{2} b x} + \frac {a \sqrt [3]{- \frac {a}{b}} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{3 a^{3} + 3 a^{2} b x} + \frac {2 \sqrt {3} a \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a^{3} + 3 a^{2} b x} - \frac {2 a \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{3 a^{3} + 3 a^{2} b x} - \frac {2 b x \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{3 a^{3} + 3 a^{2} b x} + \frac {b x \sqrt [3]{- \frac {a}{b}} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{3 a^{3} + 3 a^{2} b x} + \frac {2 \sqrt {3} b x \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a^{3} + 3 a^{2} b x} - \frac {2 b x \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{3 a^{3} + 3 a^{2} b x} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^{2/3} (a+b x)^2} \, dx=\frac {x^{\frac {1}{3}}}{a b x + a^{2}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^{2/3} (a+b x)^2} \, dx=-\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} + \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} + \frac {x^{\frac {1}{3}}}{{\left (b x + a\right )} a} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, a^{2} b} \]
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Time = 0.23 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^{2/3} (a+b x)^2} \, dx=\frac {2\,\ln \left (\frac {6\,b^{5/3}}{a^{2/3}}+\frac {6\,b^2\,x^{1/3}}{a}\right )}{3\,a^{5/3}\,b^{1/3}}+\frac {x^{1/3}}{a\,\left (a+b\,x\right )}+\frac {\ln \left (\frac {6\,b^2\,x^{1/3}}{a}+\frac {3\,b^{5/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{a^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{3\,a^{5/3}\,b^{1/3}}-\frac {\ln \left (\frac {6\,b^2\,x^{1/3}}{a}-\frac {3\,b^{5/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{a^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{3\,a^{5/3}\,b^{1/3}} \]
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