Integrand size = 13, antiderivative size = 143 \[ \int \frac {x^{2/3}}{(a+b x)^3} \, dx=-\frac {x^{2/3}}{2 b (a+b x)^2}+\frac {x^{2/3}}{3 a b (a+b x)}-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{4/3} b^{5/3}}+\frac {\log (a+b x)}{18 a^{4/3} b^{5/3}} \]
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Time = 0.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {43, 44, 58, 631, 210, 31} \[ \int \frac {x^{2/3}}{(a+b x)^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{4/3} b^{5/3}}+\frac {\log (a+b x)}{18 a^{4/3} b^{5/3}}+\frac {x^{2/3}}{3 a b (a+b x)}-\frac {x^{2/3}}{2 b (a+b x)^2} \]
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Rule 31
Rule 43
Rule 44
Rule 58
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{2/3}}{2 b (a+b x)^2}+\frac {\int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx}{3 b} \\ & = -\frac {x^{2/3}}{2 b (a+b x)^2}+\frac {x^{2/3}}{3 a b (a+b x)}+\frac {\int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{9 a b} \\ & = -\frac {x^{2/3}}{2 b (a+b x)^2}+\frac {x^{2/3}}{3 a b (a+b x)}+\frac {\log (a+b x)}{18 a^{4/3} b^{5/3}}+\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 )}{6 a b^2}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{6 a^{4/3} b^{5/3}} \\ & = -\frac {x^{2/3}}{2 b (a+b x)^2}+\frac {x^{2/3}}{3 a b (a+b x)}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{4/3} b^{5/3}}+\frac {\log (a+b x)}{18 a^{4/3} b^{5/3}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{3 a^{4/3} b^{5/3}} \\ & = -\frac {x^{2/3}}{2 b (a+b x)^2}+\frac {x^{2/3}}{3 a b (a+b x)}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{4/3} b^{5/3}}+\frac {\log (a+b x)}{18 a^{4/3} b^{5/3}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.93 \[ \int \frac {x^{2/3}}{(a+b x)^3} \, dx=\frac {-\frac {3 \sqrt [3]{a} b^{2/3} x^{2/3} (a-2 b x)}{(a+b x)^2}-2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{18 a^{4/3} b^{5/3}} \]
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Time = 0.10 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {\frac {x^{\frac {5}{3}}}{3 a}-\frac {x^{\frac {2}{3}}}{6 b}}{\left (b x +a \right )^{2}}+\frac {-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{3 b a}\) | \(132\) |
default | \(\frac {\frac {x^{\frac {5}{3}}}{3 a}-\frac {x^{\frac {2}{3}}}{6 b}}{\left (b x +a \right )^{2}}+\frac {-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{3 b a}\) | \(132\) |
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (102) = 204\).
Time = 0.25 (sec) , antiderivative size = 508, normalized size of antiderivative = 3.55 \[ \int \frac {x^{2/3}}{(a+b x)^3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} a + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{\frac {2}{3}}\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{\frac {1}{3}}}{b x + a}\right ) + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{\frac {2}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{\frac {1}{3}} - \left (-a b^{2}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (2 \, a b^{3} x - a^{2} b^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{\frac {2}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{\frac {1}{3}} - \left (-a b^{2}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (2 \, a b^{3} x - a^{2} b^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^{2/3}}{(a+b x)^3} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.07 \[ \int \frac {x^{2/3}}{(a+b x)^3} \, dx=\frac {2 \, b x^{\frac {5}{3}} - a x^{\frac {2}{3}}}{6 \, {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
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Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.04 \[ \int \frac {x^{2/3}}{(a+b x)^3} \, dx=-\frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b^{3}} + \frac {2 \, b x^{\frac {5}{3}} - a x^{\frac {2}{3}}}{6 \, {\left (b x + a\right )}^{2} a b} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.20 \[ \int \frac {x^{2/3}}{(a+b x)^3} \, dx=\frac {\frac {x^{5/3}}{3\,a}-\frac {x^{2/3}}{6\,b}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {\ln \left (\frac {1}{9\,a^{5/3}\,{\left (-b\right )}^{4/3}}+\frac {x^{1/3}}{9\,a^2\,b}\right )}{9\,a^{4/3}\,{\left (-b\right )}^{5/3}}+\frac {\ln \left (\frac {x^{1/3}}{9\,a^2\,b}+\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{36\,a^{5/3}\,{\left (-b\right )}^{4/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,{\left (-b\right )}^{5/3}}-\frac {\ln \left (\frac {x^{1/3}}{9\,a^2\,b}+\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{36\,a^{5/3}\,{\left (-b\right )}^{4/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,{\left (-b\right )}^{5/3}} \]
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