Integrand size = 29, antiderivative size = 73 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {-38+21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac {24+95 x}{726 \sqrt {2+3 x^2}}-\frac {8 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{121 \sqrt {11}} \]
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Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1661, 837, 12, 739, 212} \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=-\frac {8 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {3 x^2+2}}\right )}{121 \sqrt {11}}-\frac {38-21 x}{198 \left (3 x^2+2\right )^{3/2}}+\frac {95 x+24}{726 \sqrt {3 x^2+2}} \]
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Rule 12
Rule 212
Rule 739
Rule 837
Rule 1661
Rubi steps \begin{align*} \text {integral}& = -\frac {38-21 x}{198 \left (2+3 x^2\right )^{3/2}}-\frac {1}{18} \int \frac {-\frac {78}{11}-\frac {84 x}{11}}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac {24+95 x}{726 \sqrt {2+3 x^2}}+\frac {\int \frac {864}{11 (1+2 x) \sqrt {2+3 x^2}} \, dx}{1188} \\ & = -\frac {38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac {24+95 x}{726 \sqrt {2+3 x^2}}+\frac {8}{121} \int \frac {1}{(1+2 x) \sqrt {2+3 x^2}} \, dx \\ & = -\frac {38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac {24+95 x}{726 \sqrt {2+3 x^2}}-\frac {8}{121} \text {Subst}\left (\int \frac {1}{11-x^2} \, dx,x,\frac {4-3 x}{\sqrt {2+3 x^2}}\right ) \\ & = -\frac {38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac {24+95 x}{726 \sqrt {2+3 x^2}}-\frac {8 \tanh ^{-1}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{121 \sqrt {11}} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {-274+801 x+216 x^2+855 x^3}{2178 \left (2+3 x^2\right )^{3/2}}-\frac {8 \text {arctanh}\left (\frac {4-3 x}{\sqrt {22+33 x^2}}\right )}{121 \sqrt {11}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.56 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01
method | result | size |
trager | \(\frac {855 x^{3}+216 x^{2}+801 x -274}{2178 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) x +11 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right )}{1+2 x}\right )}{1331}\) | \(74\) |
default | \(\frac {x}{12 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {x}{12 \sqrt {3 x^{2}+2}}-\frac {2}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {1}{33 \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {x}{44 \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {23 x}{484 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}}}+\frac {4}{121 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}}}-\frac {8 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-12 x +5}}\right )}{1331}\) | \(133\) |
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Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {72 \, \sqrt {11} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\frac {\sqrt {11} \sqrt {3 \, x^{2} + 2} {\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \, {\left (855 \, x^{3} + 216 \, x^{2} + 801 \, x - 274\right )} \sqrt {3 \, x^{2} + 2}}{23958 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]
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Timed out. \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.11 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {8}{1331} \, \sqrt {11} \operatorname {arsinh}\left (\frac {\sqrt {6} x}{2 \, {\left | 2 \, x + 1 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {95 \, x}{726 \, \sqrt {3 \, x^{2} + 2}} + \frac {4}{121 \, \sqrt {3 \, x^{2} + 2}} + \frac {7 \, x}{66 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {19}{99 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.25 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {8}{1331} \, \sqrt {11} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {11} - \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {11} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {9 \, {\left ({\left (95 \, x + 24\right )} x + 89\right )} x - 274}{2178 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]
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Time = 0.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.99 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\sqrt {11}\,\left (8\,\ln \left (x+\frac {1}{2}\right )-8\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )\right )}{1331}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {21}{176}+\frac {\sqrt {6}\,19{}\mathrm {i}}{176}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {7}{88}+\frac {\sqrt {6}\,19{}\mathrm {i}}{264}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {21}{176}+\frac {\sqrt {6}\,19{}\mathrm {i}}{176}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {7}{88}+\frac {\sqrt {6}\,19{}\mathrm {i}}{264}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-288+\sqrt {6}\,303{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{104544\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (288+\sqrt {6}\,303{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{104544\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]
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