Integrand size = 21, antiderivative size = 114 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=-\frac {c \arctan \left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \arctan \left (1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac {c \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac {1}{12} d \log \left (2+3 x^4\right ) \]
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Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1607, 1845, 303, 1176, 631, 210, 1179, 642, 266} \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=-\frac {c \arctan \left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \arctan \left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac {c \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{4\ 6^{3/4}}-\frac {c \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{4\ 6^{3/4}}+\frac {1}{12} d \log \left (3 x^4+2\right ) \]
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Rule 210
Rule 266
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1607
Rule 1845
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 (c+d x)}{2+3 x^4} \, dx \\ & = \int \left (\frac {c x^2}{2+3 x^4}+\frac {d x^3}{2+3 x^4}\right ) \, dx \\ & = c \int \frac {x^2}{2+3 x^4} \, dx+d \int \frac {x^3}{2+3 x^4} \, dx \\ & = \frac {1}{12} d \log \left (2+3 x^4\right )-\frac {c \int \frac {\sqrt {2}-\sqrt {3} x^2}{2+3 x^4} \, dx}{2 \sqrt {3}}+\frac {c \int \frac {\sqrt {2}+\sqrt {3} x^2}{2+3 x^4} \, dx}{2 \sqrt {3}} \\ & = \frac {1}{12} d \log \left (2+3 x^4\right )+\frac {1}{12} c \int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {1}{12} c \int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {c \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{4\ 6^{3/4}}+\frac {c \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{4\ 6^{3/4}} \\ & = \frac {c \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac {c \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac {1}{12} d \log \left (2+3 x^4\right )+\frac {c \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}-\frac {c \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}} \\ & = -\frac {c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac {c \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac {1}{12} d \log \left (2+3 x^4\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.95 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \left (-2 \sqrt [4]{6} c \arctan \left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} c \arctan \left (1+\sqrt [4]{6} x\right )+\sqrt [4]{6} c \log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )-\sqrt [4]{6} c \log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+2 d \log \left (2+3 x^4\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.49 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.31
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{3} d +\textit {\_R}^{2} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{12}\) | \(35\) |
default | \(\frac {c \sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{144}+\frac {d \ln \left (3 x^{4}+2\right )}{12}\) | \(106\) |
meijerg | \(\frac {d \ln \left (\frac {3 x^{4}}{2}+1\right )}{12}+\frac {54^{\frac {3}{4}} c \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{216}\) | \(183\) |
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Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (83) = 166\).
Time = 0.28 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.10 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \, {\left (\sqrt {2} \sqrt {\sqrt {6} \sqrt {-c^{4}}} + 2 \, d\right )} \log \left (6 \, c^{3} x + \sqrt {6} \sqrt {2} \sqrt {-c^{4}} \sqrt {\sqrt {6} \sqrt {-c^{4}}}\right ) - \frac {1}{24} \, {\left (\sqrt {2} \sqrt {\sqrt {6} \sqrt {-c^{4}}} - 2 \, d\right )} \log \left (6 \, c^{3} x - \sqrt {6} \sqrt {2} \sqrt {-c^{4}} \sqrt {\sqrt {6} \sqrt {-c^{4}}}\right ) - \frac {1}{24} \, {\left (\sqrt {2} \sqrt {-\sqrt {6} \sqrt {-c^{4}}} - 2 \, d\right )} \log \left (6 \, c^{3} x + \sqrt {6} \sqrt {2} \sqrt {-c^{4}} \sqrt {-\sqrt {6} \sqrt {-c^{4}}}\right ) + \frac {1}{24} \, {\left (\sqrt {2} \sqrt {-\sqrt {6} \sqrt {-c^{4}}} + 2 \, d\right )} \log \left (6 \, c^{3} x - \sqrt {6} \sqrt {2} \sqrt {-c^{4}} \sqrt {-\sqrt {6} \sqrt {-c^{4}}}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.61 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\operatorname {RootSum} {\left (41472 t^{4} - 13824 t^{3} d + 1728 t^{2} d^{2} - 96 t d^{3} + 3 c^{4} + 2 d^{4}, \left ( t \mapsto t \log {\left (x + \frac {3456 t^{3} - 864 t^{2} d + 72 t d^{2} - 2 d^{3}}{3 c^{3}} \right )} \right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.33 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{72} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} d - \sqrt {3} c\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{72} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} d + \sqrt {3} c\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{12} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} c \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{12} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} c \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{12} \cdot 6^{\frac {1}{4}} c \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{12} \cdot 6^{\frac {1}{4}} c \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) - \frac {1}{24} \, {\left (6^{\frac {1}{4}} c - 2 \, d\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) + \frac {1}{24} \, {\left (6^{\frac {1}{4}} c + 2 \, d\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
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Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.03 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\ln \left (x-\frac {{\left (-1\right )}^{1/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}+\frac {6^{1/4}\,\sqrt {-\frac {1}{2}{}\mathrm {i}}\,c}{12}\right )+\ln \left (x+\frac {{\left (-1\right )}^{1/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}-\frac {6^{1/4}\,\sqrt {-\frac {1}{2}{}\mathrm {i}}\,c}{12}\right )+\ln \left (x-\frac {{\left (-1\right )}^{3/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}-\frac {6^{1/4}\,\sqrt {\frac {1}{2}{}\mathrm {i}}\,c}{12}\right )+\ln \left (x+\frac {{\left (-1\right )}^{3/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}+\frac {6^{1/4}\,\sqrt {\frac {1}{2}{}\mathrm {i}}\,c}{12}\right ) \]
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