Integrand size = 30, antiderivative size = 53 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a-b x^3} \, dx=\frac {2 C \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}}{\sqrt {3}}\right )}{\sqrt {3} b}-\frac {C \log \left (\sqrt [3]{-\frac {a}{b}}+x\right )}{b} \]
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Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1881, 31, 631, 210} \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a-b x^3} \, dx=\frac {2 C \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}}{\sqrt {3}}\right )}{\sqrt {3} b}-\frac {C \log \left (\sqrt [3]{-\frac {a}{b}}+x\right )}{b} \]
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Rule 31
Rule 210
Rule 631
Rule 1881
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. \(150\) vs. \(2(53)=106\).
Time = 0.07 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.83 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a-b x^3} \, dx=\frac {C \left (2 \sqrt {3} \left (-\frac {a}{b}\right )^{2/3} b^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \left (-\frac {a}{b}\right )^{2/3} b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+\left (-\frac {a}{b}\right )^{2/3} b^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-a^{2/3} \log \left (a-b x^3\right )\right )}{3 a^{2/3} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(118\) vs. \(2(46)=92\).
Time = 1.48 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.25
method | result | size |
default | \(C \left (2 \left (-\frac {a}{b}\right )^{\frac {2}{3}} \left (-\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (1+\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-\frac {\ln \left (-b \,x^{3}+a \right )}{3 b}\right )\) | \(119\) |
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none
Time = 0.78 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a-b x^3} \, dx=\frac {2 \, \sqrt {3} C \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) - 3 \, C \log \left (x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b} \]
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Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.08 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a-b x^3} \, dx=- \frac {C \left (\log {\left (- \frac {a}{b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )} + \frac {\sqrt {3} i \log {\left (\frac {a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\sqrt {3} i a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3} - \frac {\sqrt {3} i \log {\left (\frac {a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} i a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (46) = 92\).
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.15 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a-b x^3} \, dx=-\frac {2 \, \sqrt {3} {\left (C a - {\left (3 \, C \left (\frac {a}{b}\right )^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {2}{3}} + \frac {C a}{b}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} - \frac {{\left (C \left (\frac {a}{b}\right )^{\frac {2}{3}} - C \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )} \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (C \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, C \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )} \log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.06 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a-b x^3} \, dx=-\frac {\sqrt {3} {\left (a b^{2} - i \, \sqrt {3} \sqrt {a^{2} b^{4}}\right )} C \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{3}} - \frac {{\left (C b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} C\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{2}} \]
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Time = 9.64 (sec) , antiderivative size = 172, normalized size of antiderivative = 3.25 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a-b x^3} \, dx=\sum _{k=1}^3\ln \left (-\frac {\left (C+\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z+9\,C^3\,a^2,z,k\right )\,b\,3\right )\,\left (C\,a+\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z+9\,C^3\,a^2,z,k\right )\,a\,b\,3+2\,C\,b\,x\,{\left (-\frac {a}{b}\right )}^{2/3}\right )}{b^3}\right )\,\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z+9\,C^3\,a^2,z,k\right ) \]
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