Integrand size = 13, antiderivative size = 110 \[ \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx=-\frac {1}{a x}+\frac {-\sqrt {3} \arctan \left (\frac {-\sqrt [3]{\frac {a}{b}}+2 x}{\sqrt {3} \sqrt [3]{\frac {a}{b}}}\right )+\frac {1}{2} \log \left (\frac {\left (\sqrt [3]{\frac {a}{b}}+x\right )^2}{\left (\frac {a}{b}\right )^{2/3}-\sqrt [3]{\frac {a}{b}} x+x^2}\right )}{3 a \sqrt [3]{\frac {a}{b}}} \]
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Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {331, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx=\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3}}-\frac {1}{a x} \]
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Rule 31
Rule 210
Rule 298
Rule 331
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a x}-\frac {b \int \frac {x}{a+b x^3} \, dx}{a} \\ & = -\frac {1}{a x}+\frac {b^{2/3} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{4/3}}-\frac {b^{2/3} \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{4/3}} \\ & = -\frac {1}{a x}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3}}-\frac {\sqrt [3]{b} \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}{6 a^{4/3}}-\frac {b^{2/3} \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a} \\ & = -\frac {1}{a x}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}-\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{4/3}} \\ & = -\frac {1}{a x}+\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx=\frac {-6 \sqrt [3]{a}+2 \sqrt {3} \sqrt [3]{b} x \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{b} x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\sqrt [3]{b} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} x} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.48
method | result | size |
risch | \(-\frac {1}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{4}+3 b \right ) x -a^{3} \textit {\_R}^{2}\right )\right )}{3}\) | \(53\) |
default | \(-\frac {1}{a x}-\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b}{a}\) | \(106\) |
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Time = 0.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx=-\frac {2 \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, x \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6}{6 \, a x} \]
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Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{4} - b, \left ( t \mapsto t \log {\left (\frac {9 t^{2} a^{3}}{b} + x \right )} \right )\right )} - \frac {1}{a x} \]
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Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {1}{a x} \]
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Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx=\frac {b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b} - \frac {1}{a x} \]
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Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx=\frac {b^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )}{3\,a^{4/3}}-\frac {1}{a\,x}-\frac {b^{1/3}\,\ln \left (4\,b^{1/3}\,x-2\,a^{1/3}+\sqrt {3}\,a^{1/3}\,2{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}}+\frac {b^{1/3}\,\ln \left (4\,b^{1/3}\,x-2\,a^{1/3}-\sqrt {3}\,a^{1/3}\,2{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}} \]
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