Integrand size = 42, antiderivative size = 71 \[ \int \frac {\sqrt [3]{\frac {a}{b}} B+2 \left (\frac {a}{b}\right )^{2/3} C+B x+C x^2}{a+b x^3} \, dx=-\frac {2 \left (\frac {a}{b}\right )^{2/3} \left (B+\sqrt [3]{\frac {a}{b}} C\right ) \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {a}{b}}}}{\sqrt {3}}\right )}{\sqrt {3} a}+\frac {C \log \left (\sqrt [3]{\frac {a}{b}}+x\right )}{b} \]
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Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1881, 31, 631, 210} \[ \int \frac {\sqrt [3]{\frac {a}{b}} B+2 \left (\frac {a}{b}\right )^{2/3} C+B x+C x^2}{a+b x^3} \, dx=\frac {C \log \left (\sqrt [3]{\frac {a}{b}}+x\right )}{b}-\frac {2 \left (\frac {a}{b}\right )^{2/3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {a}{b}}}}{\sqrt {3}}\right ) \left (C \sqrt [3]{\frac {a}{b}}+B\right )}{\sqrt {3} a} \]
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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 (B+\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}+\left (2 \left (\frac {\left (\frac {a}{b}\right )^{2/3} B}{a}+\frac {C}{b}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{\frac {a}{b}}}\right ) \\ & = -\frac {2 \left (\frac {\left (\frac {a}{b}\right )^{2/3} B}{a}+\frac {C}{b}\right ) \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {a}{b}}}}{\sqrt {3}}\right )}{\sqrt {3}}+\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. \(247\) vs. \(2(71)=142\).
Time = 0.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.48 \[ \int \frac {\sqrt [3]{\frac {a}{b}} B+2 \left (\frac {a}{b}\right )^{2/3} C+B x+C x^2}{a+b x^3} \, dx=\frac {2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} B+\sqrt [3]{\frac {a}{b}} \sqrt [3]{b} \left (B+2 \sqrt [3]{\frac {a}{b}} C\right )\right ) \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )+2 \sqrt [3]{b} \left (-a^{2/3} B+\sqrt [3]{a} \sqrt [3]{\frac {a}{b}} \sqrt [3]{b} \left (B+2 \sqrt [3]{\frac {a}{b}} C\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt [3]{b} \left (a^{2/3} B-\sqrt [3]{a} \sqrt [3]{\frac {a}{b}} \sqrt [3]{b} \left (B+2 \sqrt [3]{\frac {a}{b}} C\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 a C \log \left (a+b x^3\right )}{6 a b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(218\) vs. \(2(60)=120\).
Time = 1.50 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.08
method | result | size |
default | \(\left (2 \left (\frac {a}{b}\right )^{\frac {2}{3}} C +\left (\frac {a}{b}\right )^{\frac {1}{3}} B \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+B \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {C \ln \left (b \,x^{3}+a \right )}{3 b}\) | \(219\) |
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Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (60) = 120\).
Time = 1.98 (sec) , antiderivative size = 429, normalized size of antiderivative = 6.04 \[ \int \frac {\sqrt [3]{\frac {a}{b}} B+2 \left (\frac {a}{b}\right )^{2/3} C+B x+C x^2}{a+b x^3} \, dx=\left [\frac {C \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + \sqrt {\frac {1}{3}} \sqrt {-\frac {2 \, B C b \left (\frac {a}{b}\right )^{\frac {2}{3}} + B^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}} + C^{2} a}{a}} \log \left (-\frac {C^{3} a^{2} + B^{3} a b - 2 \, {\left (C^{3} a b + B^{3} b^{2}\right )} x^{3} + 3 \, {\left (C^{3} a b + B^{3} b^{2}\right )} x \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, B C a b x^{2} - B^{2} a b x + C^{2} a^{2} - {\left (2 \, B^{2} b^{2} x^{2} + C^{2} a b x + B C a b\right )} \left (\frac {a}{b}\right )^{\frac {2}{3}} - {\left (2 \, C^{2} a b x^{2} - B C a b x - B^{2} a b\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {2 \, B C b \left (\frac {a}{b}\right )^{\frac {2}{3}} + B^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}} + C^{2} a}{a}}}{b x^{3} + a}\right )}{b}, \frac {2 \, \sqrt {\frac {1}{3}} \sqrt {\frac {2 \, B C b \left (\frac {a}{b}\right )^{\frac {2}{3}} + B^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}} + C^{2} a}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, B^{2} b x - C^{2} a + {\left (2 \, C^{2} b x + B C b\right )} \left (\frac {a}{b}\right )^{\frac {2}{3}} - {\left (2 \, B C b x + B^{2} b\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \sqrt {\frac {2 \, B C b \left (\frac {a}{b}\right )^{\frac {2}{3}} + B^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}} + C^{2} a}{a}}}{C^{3} a + B^{3} b}\right ) + C \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b}\right ] \]
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Timed out. \[ \int \frac {\sqrt [3]{\frac {a}{b}} B+2 \left (\frac {a}{b}\right )^{2/3} C+B x+C x^2}{a+b x^3} \, dx=\text {Timed out} \]
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none
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt [3]{\frac {a}{b}} B+2 \left (\frac {a}{b}\right )^{2/3} C+B x+C x^2}{a+b x^3} \, dx=\frac {C \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b} - \frac {2 \, \sqrt {3} {\left (C a - {\left (3 \, B \left (\frac {a}{b}\right )^{\frac {2}{3}} + \frac {4 \, C a}{b}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.73 \[ \int \frac {\sqrt [3]{\frac {a}{b}} B+2 \left (\frac {a}{b}\right )^{2/3} C+B x+C x^2}{a+b x^3} \, dx=\frac {{\left (2 \, C a b + \left (-a^{2} b^{4}\right )^{\frac {1}{3}} B\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, a b^{2} - i \, \sqrt {3} \sqrt {a^{2} b^{4}}} - \frac {2 \, \sqrt {3} {\left (C a b + \left (a b^{2}\right )^{\frac {2}{3}} B\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{2}} - \frac {{\left (C b^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} + B b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (a b^{2}\right )^{\frac {1}{3}} B b + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} C\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{2}} \]
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Time = 10.86 (sec) , antiderivative size = 436, normalized size of antiderivative = 6.14 \[ \int \frac {\sqrt [3]{\frac {a}{b}} B+2 \left (\frac {a}{b}\right )^{2/3} C+B x+C x^2}{a+b x^3} \, dx=\sum _{k=1}^3\ln \left (\frac {C^2\,a+B^2\,b\,{\left (\frac {a}{b}\right )}^{1/3}+2\,B\,C\,b\,{\left (\frac {a}{b}\right )}^{2/3}}{b^3}+\frac {\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+18\,B\,C\,a\,b^2\,z\,{\left (\frac {a}{b}\right )}^{2/3}+9\,B^2\,a\,b^2\,z\,{\left (\frac {a}{b}\right )}^{1/3}+9\,C^2\,a^2\,b\,z-18\,B\,C^2\,a\,b\,{\left (\frac {a}{b}\right )}^{2/3}-9\,B^2\,C\,a\,b\,{\left (\frac {a}{b}\right )}^{1/3}-9\,C^3\,a^2,z,k\right )\,\left (-6\,C\,a+\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+18\,B\,C\,a\,b^2\,z\,{\left (\frac {a}{b}\right )}^{2/3}+9\,B^2\,a\,b^2\,z\,{\left (\frac {a}{b}\right )}^{1/3}+9\,C^2\,a^2\,b\,z-18\,B\,C^2\,a\,b\,{\left (\frac {a}{b}\right )}^{2/3}-9\,B^2\,C\,a\,b\,{\left (\frac {a}{b}\right )}^{1/3}-9\,C^3\,a^2,z,k\right )\,a\,b\,9+3\,B\,b\,x\,{\left (\frac {a}{b}\right )}^{1/3}+6\,C\,b\,x\,{\left (\frac {a}{b}\right )}^{2/3}\right )}{b^2}-\frac {x\,\left (2\,C^2\,{\left (\frac {a}{b}\right )}^{2/3}-B^2+B\,C\,{\left (\frac {a}{b}\right )}^{1/3}\right )}{b^2}\right )\,\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+18\,B\,C\,a\,b^2\,z\,{\left (\frac {a}{b}\right )}^{2/3}+9\,B^2\,a\,b^2\,z\,{\left (\frac {a}{b}\right )}^{1/3}+9\,C^2\,a^2\,b\,z-18\,B\,C^2\,a\,b\,{\left (\frac {a}{b}\right )}^{2/3}-9\,B^2\,C\,a\,b\,{\left (\frac {a}{b}\right )}^{1/3}-9\,C^3\,a^2,z,k\right ) \]
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