Integrand size = 13, antiderivative size = 126 \[ \int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx=\frac {-\frac {1}{2 a x^2}-\frac {5 b x}{6 a^2}}{a+b x^3}-\frac {5 \sqrt [3]{\frac {a}{b}} b \left (\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{2 \sqrt [3]{\frac {a}{b}}-x}\right )+\frac {1}{2} \log \left (\frac {\left (\sqrt [3]{\frac {a}{b}}+x\right )^2}{\left (\frac {a}{b}\right )^{2/3}-\sqrt [3]{\frac {a}{b}} x+x^2}\right )\right )}{9 a^3} \]
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Time = 0.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {296, 331, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx=\frac {5 b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3}}+\frac {5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}-\frac {5}{6 a^2 x^2}+\frac {1}{3 a x^2 \left (a+b x^3\right )} \]
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Rule 31
Rule 206
Rule 210
Rule 296
Rule 331
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3 a x^2 \left (a+b x^3\right )}+\frac {5 \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx}{3 a} \\ & = -\frac {5}{6 a^2 x^2}+\frac {1}{3 a x^2 \left (a+b x^3\right )}-\frac {(5 b) \int \frac {1}{a+b x^3} \, dx}{3 a^2} \\ & = -\frac {5}{6 a^2 x^2}+\frac {1}{3 a x^2 \left (a+b x^3\right )}-\frac {(5 b) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{8/3}}-\frac {(5 b) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{8/3}} \\ & = -\frac {5}{6 a^2 x^2}+\frac {1}{3 a x^2 \left (a+b x^3\right )}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {\left (5 b^{2/3}\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{8/3}}-\frac {(5 b) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{7/3}} \\ & = -\frac {5}{6 a^2 x^2}+\frac {1}{3 a x^2 \left (a+b x^3\right )}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}-\frac {\left (5 b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{8/3}} \\ & = -\frac {5}{6 a^2 x^2}+\frac {1}{3 a x^2 \left (a+b x^3\right )}+\frac {5 b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3}}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx=\frac {-\frac {9 a^{2/3}}{x^2}-\frac {6 a^{2/3} b x}{a+b x^3}+10 \sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-10 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {-\frac {5 b \,x^{3}}{6 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b \,x^{3}+a \right )}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{8} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{8}-3 b^{2}\right ) x -a^{3} b \textit {\_R} \right )\right )}{9}\) | \(74\) |
default | \(-\frac {b \left (\frac {x}{3 b \,x^{3}+3 a}+\frac {5 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {5 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{a^{2}}-\frac {1}{2 a^{2} x^{2}}\) | \(118\) |
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Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx=-\frac {15 \, b x^{3} - 10 \, \sqrt {3} {\left (b x^{5} + a x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 5 \, {\left (b x^{5} + a x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 10 \, {\left (b x^{5} + a x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 9 \, a}{18 \, {\left (a^{2} b x^{5} + a^{3} x^{2}\right )}} \]
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Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.46 \[ \int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx=\frac {- 3 a - 5 b x^{3}}{6 a^{3} x^{2} + 6 a^{2} b x^{5}} + \operatorname {RootSum} {\left (729 t^{3} a^{8} + 125 b^{2}, \left ( t \mapsto t \log {\left (- \frac {9 t a^{3}}{5 b} + x \right )} \right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx=-\frac {5 \, b x^{3} + 3 \, a}{6 \, {\left (a^{2} b x^{5} + a^{3} x^{2}\right )}} - \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx=\frac {5 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3}} - \frac {b x}{3 \, {\left (b x^{3} + a\right )} a^{2}} - \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3}} - \frac {5 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3}} - \frac {1}{2 \, a^{2} x^{2}} \]
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Time = 14.67 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx=\frac {5\,{\left (-1\right )}^{1/3}\,b^{2/3}\,\ln \left ({\left (-1\right )}^{1/3}\,a^{1/3}-b^{1/3}\,x\right )}{9\,a^{8/3}}-\frac {\frac {1}{2\,a}+\frac {5\,b\,x^3}{6\,a^2}}{b\,x^5+a\,x^2}-\frac {5\,{\left (-1\right )}^{1/3}\,b^{2/3}\,\ln \left ({\left (-1\right )}^{1/3}\,a^{1/3}+2\,b^{1/3}\,x+{\left (-1\right )}^{5/6}\,\sqrt {3}\,a^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{8/3}}+\frac {5\,{\left (-1\right )}^{1/3}\,b^{2/3}\,\ln \left ({\left (-1\right )}^{1/3}\,a^{1/3}+2\,b^{1/3}\,x-{\left (-1\right )}^{5/6}\,\sqrt {3}\,a^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{8/3}} \]
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