Integrand size = 11, antiderivative size = 49 \[ \int \frac {1}{\left (3+2 x^2\right )^{7/2}} \, dx=\frac {x}{15 \left (3+2 x^2\right )^{5/2}}+\frac {4 x}{135 \left (3+2 x^2\right )^{3/2}}+\frac {8 x}{405 \sqrt {3+2 x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {198, 197} \[ \int \frac {1}{\left (3+2 x^2\right )^{7/2}} \, dx=\frac {8 x}{405 \sqrt {2 x^2+3}}+\frac {4 x}{135 \left (2 x^2+3\right )^{3/2}}+\frac {x}{15 \left (2 x^2+3\right )^{5/2}} \]
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Rule 197
Rule 198
Rubi steps \begin{align*} \text {integral}& = \frac {x}{15 \left (3+2 x^2\right )^{5/2}}+\frac {4}{15} \int \frac {1}{\left (3+2 x^2\right )^{5/2}} \, dx \\ & = \frac {x}{15 \left (3+2 x^2\right )^{5/2}}+\frac {4 x}{135 \left (3+2 x^2\right )^{3/2}}+\frac {8}{135} \int \frac {1}{\left (3+2 x^2\right )^{3/2}} \, dx \\ & = \frac {x}{15 \left (3+2 x^2\right )^{5/2}}+\frac {4 x}{135 \left (3+2 x^2\right )^{3/2}}+\frac {8 x}{405 \sqrt {3+2 x^2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (3+2 x^2\right )^{7/2}} \, dx=\frac {x \left (135+120 x^2+32 x^4\right )}{405 \left (3+2 x^2\right )^{5/2}} \]
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Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(\frac {x \left (32 x^{4}+120 x^{2}+135\right )}{405 \left (2 x^{2}+3\right )^{\frac {5}{2}}}\) | \(25\) |
trager | \(\frac {x \left (32 x^{4}+120 x^{2}+135\right )}{405 \left (2 x^{2}+3\right )^{\frac {5}{2}}}\) | \(25\) |
risch | \(\frac {x \left (32 x^{4}+120 x^{2}+135\right )}{405 \left (2 x^{2}+3\right )^{\frac {5}{2}}}\) | \(25\) |
pseudoelliptic | \(\frac {32 x^{5}+120 x^{3}+135 x}{405 \left (2 x^{2}+3\right )^{\frac {5}{2}}}\) | \(26\) |
meijerg | \(\frac {\sqrt {3}\, x \left (\frac {32}{9} x^{4}+\frac {40}{3} x^{2}+15\right )}{1215 \left (1+\frac {2 x^{2}}{3}\right )^{\frac {5}{2}}}\) | \(28\) |
default | \(\frac {x}{15 \left (2 x^{2}+3\right )^{\frac {5}{2}}}+\frac {4 x}{135 \left (2 x^{2}+3\right )^{\frac {3}{2}}}+\frac {8 x}{405 \sqrt {2 x^{2}+3}}\) | \(38\) |
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Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (3+2 x^2\right )^{7/2}} \, dx=\frac {{\left (32 \, x^{5} + 120 \, x^{3} + 135 \, x\right )} \sqrt {2 \, x^{2} + 3}}{405 \, {\left (8 \, x^{6} + 36 \, x^{4} + 54 \, x^{2} + 27\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (42) = 84\).
Time = 2.93 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.84 \[ \int \frac {1}{\left (3+2 x^2\right )^{7/2}} \, dx=\frac {32 x^{5}}{1620 x^{4} \sqrt {2 x^{2} + 3} + 4860 x^{2} \sqrt {2 x^{2} + 3} + 3645 \sqrt {2 x^{2} + 3}} + \frac {120 x^{3}}{1620 x^{4} \sqrt {2 x^{2} + 3} + 4860 x^{2} \sqrt {2 x^{2} + 3} + 3645 \sqrt {2 x^{2} + 3}} + \frac {135 x}{1620 x^{4} \sqrt {2 x^{2} + 3} + 4860 x^{2} \sqrt {2 x^{2} + 3} + 3645 \sqrt {2 x^{2} + 3}} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (3+2 x^2\right )^{7/2}} \, dx=\frac {8 \, x}{405 \, \sqrt {2 \, x^{2} + 3}} + \frac {4 \, x}{135 \, {\left (2 \, x^{2} + 3\right )}^{\frac {3}{2}}} + \frac {x}{15 \, {\left (2 \, x^{2} + 3\right )}^{\frac {5}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (3+2 x^2\right )^{7/2}} \, dx=\frac {{\left (8 \, {\left (4 \, x^{2} + 15\right )} x^{2} + 135\right )} x}{405 \, {\left (2 \, x^{2} + 3\right )}^{\frac {5}{2}}} \]
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Time = 0.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 3.82 \[ \int \frac {1}{\left (3+2 x^2\right )^{7/2}} \, dx=\frac {2\,\sqrt {2}\,\sqrt {x^2+\frac {3}{2}}}{405\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{2}\right )}+\frac {2\,\sqrt {2}\,\sqrt {x^2+\frac {3}{2}}}{405\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{2}\right )}+\frac {\sqrt {2}\,\sqrt {x^2+\frac {3}{2}}}{1440\,\left (-x^3+\frac {3{}\mathrm {i}\,\sqrt {6}\,x^2}{2}+\frac {9\,x}{2}-\frac {\sqrt {6}\,3{}\mathrm {i}}{4}\right )}+\frac {\sqrt {2}\,\sqrt {x^2+\frac {3}{2}}}{1440\,\left (-x^3-\frac {3{}\mathrm {i}\,\sqrt {6}\,x^2}{2}+\frac {9\,x}{2}+\frac {\sqrt {6}\,3{}\mathrm {i}}{4}\right )}+\frac {\sqrt {2}\,\sqrt {6}\,\sqrt {x^2+\frac {3}{2}}\,19{}\mathrm {i}}{25920\,\left (x^2+1{}\mathrm {i}\,\sqrt {6}\,x-\frac {3}{2}\right )}+\frac {\sqrt {2}\,\sqrt {6}\,\sqrt {x^2+\frac {3}{2}}\,19{}\mathrm {i}}{25920\,\left (-x^2+1{}\mathrm {i}\,\sqrt {6}\,x+\frac {3}{2}\right )} \]
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