Integrand size = 13, antiderivative size = 140 \[ \int \frac {x^{5/3}}{(a+b x)^3} \, dx=-\frac {x^{5/3}}{2 b (a+b x)^2}-\frac {5 x^{2/3}}{6 b^2 (a+b x)}-\frac {5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{8/3}}-\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 \sqrt [3]{a} b^{8/3}}+\frac {5 \log (a+b x)}{18 \sqrt [3]{a} b^{8/3}} \]
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Time = 0.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {43, 58, 631, 210, 31} \[ \int \frac {x^{5/3}}{(a+b x)^3} \, dx=-\frac {5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{8/3}}-\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 \sqrt [3]{a} b^{8/3}}+\frac {5 \log (a+b x)}{18 \sqrt [3]{a} b^{8/3}}-\frac {5 x^{2/3}}{6 b^2 (a+b x)}-\frac {x^{5/3}}{2 b (a+b x)^2} \]
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Rule 31
Rule 43
Rule 58
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{5/3}}{2 b (a+b x)^2}+\frac {5 \int \frac {x^{2/3}}{(a+b x)^2} \, dx}{6 b} \\ & = -\frac {x^{5/3}}{2 b (a+b x)^2}-\frac {5 x^{2/3}}{6 b^2 (a+b x)}+\frac {5 \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{9 b^2} \\ & = -\frac {x^{5/3}}{2 b (a+b x)^2}-\frac {5 x^{2/3}}{6 b^2 (a+b x)}+\frac {5 \log (a+b x)}{18 \sqrt [3]{a} b^{8/3}}+\frac {5 \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 b^3}-\frac {5 \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{a} b^{8/3}} \\ & = -\frac {x^{5/3}}{2 b (a+b x)^2}-\frac {5 x^{2/3}}{6 b^2 (a+b x)}-\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 \sqrt [3]{a} b^{8/3}}+\frac {5 \log (a+b x)}{18 \sqrt [3]{a} b^{8/3}}+\frac {5 \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 \sqrt [3]{a} b^{8/3}} \\ & = -\frac {x^{5/3}}{2 b (a+b x)^2}-\frac {5 x^{2/3}}{6 b^2 (a+b x)}-\frac {5 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{8/3}}-\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 \sqrt [3]{a} b^{8/3}}+\frac {5 \log (a+b x)}{18 \sqrt [3]{a} b^{8/3}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int \frac {x^{5/3}}{(a+b x)^3} \, dx=\frac {-\frac {3 b^{2/3} x^{2/3} (5 a+8 b x)}{(a+b x)^2}-\frac {10 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{a}}+\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{\sqrt [3]{a}}}{18 b^{8/3}} \]
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Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {-\frac {4 x^{\frac {5}{3}}}{3 b}-\frac {5 a \,x^{\frac {2}{3}}}{6 b^{2}}}{\left (b x +a \right )^{2}}+\frac {-\frac {5 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{b^{2}}\) | \(130\) |
default | \(\frac {-\frac {4 x^{\frac {5}{3}}}{3 b}-\frac {5 a \,x^{\frac {2}{3}}}{6 b^{2}}}{\left (b x +a \right )^{2}}+\frac {-\frac {5 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{b^{2}}\) | \(130\) |
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Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (99) = 198\).
Time = 0.24 (sec) , antiderivative size = 506, normalized size of antiderivative = 3.61 \[ \int \frac {x^{5/3}}{(a+b x)^3} \, dx=\left [\frac {15 \, \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 ) + 5 \, {\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 ) - 10 \, {\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 (8 \, a b^{3} x + 5 \, a^{2} b^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a b^{6} x^{2} + 2 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}, \frac {30 \, \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 ) + 5 \, {\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 ) - 10 \, {\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 (8 \, a b^{3} x + 5 \, a^{2} b^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a b^{6} x^{2} + 2 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^{5/3}}{(a+b x)^3} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.02 \[ \int \frac {x^{5/3}}{(a+b x)^3} \, dx=-\frac {8 \, b x^{\frac {5}{3}} + 5 \, a x^{\frac {2}{3}}}{6 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} + \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {5 \, \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {5 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
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Time = 0.30 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04 \[ \int \frac {x^{5/3}}{(a+b x)^3} \, dx=-\frac {5 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{2}} - \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{4}} - \frac {8 \, b x^{\frac {5}{3}} + 5 \, a x^{\frac {2}{3}}}{6 \, {\left (b x + a\right )}^{2} b^{2}} + \frac {5 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.18 \[ \int \frac {x^{5/3}}{(a+b x)^3} \, dx=\frac {5\,\ln \left (\frac {25\,x^{1/3}}{9\,b^3}-\frac {25\,{\left (-a\right )}^{1/3}}{9\,b^{10/3}}\right )}{9\,{\left (-a\right )}^{1/3}\,b^{8/3}}-\frac {\frac {4\,x^{5/3}}{3\,b}+\frac {5\,a\,x^{2/3}}{6\,b^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {\ln \left (\frac {25\,x^{1/3}}{9\,b^3}-\frac {{\left (-a\right )}^{1/3}\,{\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}^2}{36\,b^{10/3}}\right )\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{18\,{\left (-a\right )}^{1/3}\,b^{8/3}}-\frac {\ln \left (\frac {25\,x^{1/3}}{9\,b^3}-\frac {{\left (-a\right )}^{1/3}\,{\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}^2}{36\,b^{10/3}}\right )\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{18\,{\left (-a\right )}^{1/3}\,b^{8/3}} \]
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