Integrand size = 32, antiderivative size = 70 \[ \int \frac {-2 a^{2/3} C-(-b)^{2/3} C x^2}{a+b x^3} \, dx=-\frac {2 C \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{-b}}+\frac {C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}} \]
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Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1880, 31, 631, 210} \[ \int \frac {-2 a^{2/3} C-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\frac {C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}-\frac {2 C \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{-b}} \]
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Rule 31
Rule 210
Rule 631
Rule 1880
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\sqrt [3]{a} C\right ) \int \frac {1}{\frac {a^{2/3}}{(-b)^{2/3}}+\frac {\sqrt [3]{a} x}{\sqrt [3]{-b}}+x^2} \, dx}{(-b)^{2/3}}-\frac {C \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{-b}}-x} \, dx}{\sqrt [3]{-b}} \\ & = \frac {C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}+\frac {(2 C) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{-b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{-b}} \\ & = -\frac {2 C \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{-b}}+\frac {C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.66 \[ \int \frac {-2 a^{2/3} C-(-b)^{2/3} C x^2}{a+b x^3} \, dx=-\frac {C \left (-2 \sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+(-b)^{2/3} \log \left (a+b x^3\right )\right )}{3 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(116\) vs. \(2(53)=106\).
Time = 1.57 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.67
method | result | size |
default | \(C \left (-2 a^{\frac {2}{3}} \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-\frac {\left (-b \right )^{\frac {2}{3}} \ln \left (b \,x^{3}+a \right )}{3 b}\right )\) | \(117\) |
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Time = 0.36 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.93 \[ \int \frac {-2 a^{2/3} C-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\left [\frac {\sqrt {\frac {1}{3}} C b \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, b x^{3} + 3 \, a^{\frac {2}{3}} \left (-b\right )^{\frac {1}{3}} x - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a^{\frac {1}{3}} b x^{2} + a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x + a \left (-b\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} - a}{b x^{3} + a}\right ) - C \left (-b\right )^{\frac {2}{3}} \log \left (b x + a^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}}\right )}{b}, -\frac {2 \, \sqrt {\frac {1}{3}} C b \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x + a \left (-b\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{a}\right ) + C \left (-b\right )^{\frac {2}{3}} \log \left (b x + a^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}}\right )}{b}\right ] \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.04 \[ \int \frac {-2 a^{2/3} C-(-b)^{2/3} C x^2}{a+b x^3} \, dx=- \operatorname {RootSum} {\left (3 t^{3} b^{2} - 3 t^{2} C b \left (- b\right )^{\frac {2}{3}} + t C^{2} \left (- b\right )^{\frac {4}{3}} - C^{3} b, \left ( t \mapsto t \log {\left (\frac {3 t \sqrt [3]{a}}{2 C} - \frac {\sqrt [3]{a} \left (- b\right )^{\frac {2}{3}}}{2 b} + x \right )} \right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (53) = 106\).
Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.47 \[ \int \frac {-2 a^{2/3} C-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\frac {2 \, \sqrt {3} {\left (C a \left (-b\right )^{\frac {2}{3}} - {\left (3 \, C a^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {C a \left (-b\right )^{\frac {2}{3}}}{b}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} - \frac {{\left (C \left (-b\right )^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} - C a^{\frac {2}{3}}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (C \left (-b\right )^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, C a^{\frac {2}{3}}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Timed out. \[ \int \frac {-2 a^{2/3} C-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\text {Timed out} \]
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Time = 9.43 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.16 \[ \int \frac {-2 a^{2/3} C-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,{\left (-b\right )}^{8/3}\,z^2-9\,C^2\,a^2\,{\left (-b\right )}^{7/3}\,z+9\,C^3\,a^2\,b^2,z,k\right )\,\left (\frac {6\,C\,a}{{\left (-b\right )}^{4/3}}+\frac {\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,{\left (-b\right )}^{8/3}\,z^2-9\,C^2\,a^2\,{\left (-b\right )}^{7/3}\,z+9\,C^3\,a^2\,b^2,z,k\right )\,a\,9}{b}-\frac {6\,C\,a^{2/3}\,x}{b}\right )-\frac {C^2\,a}{{\left (-b\right )}^{5/3}}-\frac {2\,C^2\,a^{2/3}\,x}{{\left (-b\right )}^{4/3}}\right )\,\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,{\left (-b\right )}^{8/3}\,z^2-9\,C^2\,a^2\,{\left (-b\right )}^{7/3}\,z+9\,C^3\,a^2\,b^2,z,k\right ) \]
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