Integrand size = 28, antiderivative size = 54 \[ \int \frac {x \left (2 \sqrt [3]{-\frac {a}{b}} C+C x\right )}{a+b x^3} \, dx=\frac {2 C \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}}{\sqrt {3}}\right )}{\sqrt {3} b}+\frac {C \log \left (\sqrt [3]{-\frac {a}{b}}-x\right )}{b} \]
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Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1883, 31, 631, 210} \[ \int \frac {x \left (2 \sqrt [3]{-\frac {a}{b}} C+C x\right )}{a+b x^3} \, dx=\frac {2 C \arctan \left (\frac {\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}+1}{\sqrt {3}}\right )}{\sqrt {3} b}+\frac {C \log \left (\sqrt [3]{-\frac {a}{b}}-x\right )}{b} \]
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Rule 31
Rule 210
Rule 631
Rule 1883
Rubi steps \begin{align*} \text {integral}& = -\frac {C \int \frac {1}{\sqrt [3]{-\frac {a}{b}}-x} \, dx}{b}+\frac {\left (\sqrt [3]{-\frac {a}{b}} C\right ) \int \frac {1}{\left (-\frac {a}{b}\right )^{2/3}+\sqrt [3]{-\frac {a}{b}} x+x^2} \, dx}{b} \\ & = \frac {C \log \left (\sqrt [3]{-\frac {a}{b}}-x\right )}{b}-\frac {(2 C) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}\right )}{b} \\ & = \frac {2 C \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}}{\sqrt {3}}\right )}{\sqrt {3} b}+\frac {C \log \left (\sqrt [3]{-\frac {a}{b}}-x\right )}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(148\) vs. \(2(54)=108\).
Time = 0.05 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.74 \[ \int \frac {x \left (2 \sqrt [3]{-\frac {a}{b}} C+C x\right )}{a+b x^3} \, dx=\frac {C \left (-2 \sqrt {3} \sqrt [3]{-\frac {a}{b}} \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \sqrt [3]{-\frac {a}{b}} \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt [3]{-\frac {a}{b}} \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\sqrt [3]{a} \log \left (a+b x^3\right )\right )}{3 \sqrt [3]{a} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(116\) vs. \(2(47)=94\).
Time = 1.52 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.17
method | result | size |
default | \(C \left (2 \left (-\frac {a}{b}\right )^{\frac {1}{3}} \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\ln \left (b \,x^{3}+a \right )}{3 b}\right )\) | \(117\) |
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none
Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \frac {x \left (2 \sqrt [3]{-\frac {a}{b}} C+C x\right )}{a+b x^3} \, dx=-\frac {2 \, \sqrt {3} C \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - 3 \, C \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b} \]
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Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.02 \[ \int \frac {x \left (2 \sqrt [3]{-\frac {a}{b}} C+C x\right )}{a+b x^3} \, dx=\frac {C \left (\log {\left (\frac {a}{b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )} + \frac {\sqrt {3} i \log {\left (- \frac {a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\sqrt {3} i a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3} - \frac {\sqrt {3} i \log {\left (- \frac {a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} i a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (47) = 94\).
Time = 0.35 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.09 \[ \int \frac {x \left (2 \sqrt [3]{-\frac {a}{b}} C+C x\right )}{a+b x^3} \, dx=\frac {{\left (C \left (\frac {a}{b}\right )^{\frac {1}{3}} + C \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (C \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, C \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \sqrt {3} {\left (C a - {\left (3 \, C \left (\frac {a}{b}\right )^{\frac {2}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \frac {C a}{b}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (47) = 94\).
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.80 \[ \int \frac {x \left (2 \sqrt [3]{-\frac {a}{b}} C+C x\right )}{a+b x^3} \, dx=\frac {2 \, \sqrt {3} C \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b} - \frac {{\left (C b \left (-\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} C \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b} \]
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Time = 11.06 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.87 \[ \int \frac {x \left (2 \sqrt [3]{-\frac {a}{b}} C+C x\right )}{a+b x^3} \, dx=\sum _{k=1}^3\ln \left (\frac {C^2\,a+{\mathrm {root}\left (27\,a\,b^3\,z^3-27\,C\,a\,b^2\,z^2+9\,C^2\,a\,b\,z-9\,C^3\,a,z,k\right )}^2\,a\,b^2\,9-C\,\mathrm {root}\left (27\,a\,b^3\,z^3-27\,C\,a\,b^2\,z^2+9\,C^2\,a\,b\,z-9\,C^3\,a,z,k\right )\,a\,b\,6+4\,C^2\,b\,x\,{\left (-\frac {a}{b}\right )}^{2/3}}{b^3}\right )\,\mathrm {root}\left (27\,a\,b^3\,z^3-27\,C\,a\,b^2\,z^2+9\,C^2\,a\,b\,z-9\,C^3\,a,z,k\right ) \]
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