Integrand size = 28, antiderivative size = 50 \[ \int \frac {2 \left (\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=-\frac {2 C \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {a}{b}}}}{\sqrt {3}}\right )}{\sqrt {3} b}+\frac {C \log \left (\sqrt [3]{\frac {a}{b}}+x\right )}{b} \]
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Time = 0.05 (sec) , antiderivative size = 50, 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 = {1881, 31, 631, 210} \[ \int \frac {2 \left (\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\frac {C \log \left (\sqrt [3]{\frac {a}{b}}+x\right )}{b}-\frac {2 C \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {a}{b}}}}{\sqrt {3}}\right )}{\sqrt {3} b} \]
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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. \(146\) vs. \(2(50)=100\).
Time = 0.06 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.92 \[ \int \frac {2 \left (\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\frac {C \left (-2 \sqrt {3} \left (\frac {a}{b}\right )^{2/3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \left (\frac {a}{b}\right )^{2/3} b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\left (\frac {a}{b}\right )^{2/3} b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+a^{2/3} \log \left (a+b x^3\right )\right )}{3 a^{2/3} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(115\) vs. \(2(43)=86\).
Time = 1.51 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.32
method | result | size |
default | \(C \left (2 \left (\frac {a}{b}\right )^{\frac {2}{3}} \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\frac {\ln \left (b \,x^{3}+a \right )}{3 b}\right )\) | \(116\) |
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Time = 0.47 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.04 \[ \int \frac {2 \left (\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\frac {2 \, \sqrt {3} C \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 3 \, C \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b} \]
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Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.00 \[ \int \frac {2 \left (\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\frac {C \left (\log {\left (\frac {a}{b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + x \right )} - \frac {\sqrt {3} i \log {\left (- \frac {a}{2 b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\sqrt {3} i a}{2 b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3} + \frac {\sqrt {3} i \log {\left (- \frac {a}{2 b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} i a}{2 b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3}\right )}{b} \]
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.02 \[ \int \frac {2 \left (\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\frac {2 \, \sqrt {3} C \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b} + \frac {C \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b} \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.86 \[ \int \frac {2 \left (\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=-\frac {2 \, \sqrt {3} C \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b} - \frac {{\left (C b^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} C\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{2}} + \frac {{\left (3 \, a b^{2} + i \, \sqrt {3} \sqrt {a^{2} b^{4}}\right )} C \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b^{3}} \]
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Time = 9.57 (sec) , antiderivative size = 172, normalized size of antiderivative = 3.44 \[ \int \frac {2 \left (\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\sum _{k=1}^3\ln \left (-\frac {\left (C-\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z-9\,C^3\,a^2,z,k\right )\,b\,3\right )\,\left (-C\,a+\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z-9\,C^3\,a^2,z,k\right )\,a\,b\,3+2\,C\,b\,x\,{\left (\frac {a}{b}\right )}^{2/3}\right )}{b^3}\right )\,\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z-9\,C^3\,a^2,z,k\right ) \]
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