Integrand size = 19, antiderivative size = 50 \[ \int \frac {x^4}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx=\frac {x^5}{7 \sqrt {10} \left (\sqrt {10}-x^2\right )^{7/2}}+\frac {x^5}{175 \left (\sqrt {10}-x^2\right )^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {277, 270} \[ \int \frac {x^4}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx=\frac {x^5}{5 \sqrt {10} \left (\sqrt {10}-x^2\right )^{7/2}}-\frac {x^7}{175 \left (\sqrt {10}-x^2\right )^{7/2}} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = \frac {x^5}{5 \sqrt {10} \left (\sqrt {10}-x^2\right )^{7/2}}-\frac {1}{5} \sqrt {\frac {2}{5}} \int \frac {x^6}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx \\ & = \frac {x^5}{5 \sqrt {10} \left (\sqrt {10}-x^2\right )^{7/2}}-\frac {x^7}{175 \left (\sqrt {10}-x^2\right )^{7/2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.70 \[ \int \frac {x^4}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx=-\frac {x^5 \left (-7 \sqrt {10}+2 x^2\right )}{350 \left (\sqrt {10}-x^2\right )^{7/2}} \]
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Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.56
method | result | size |
gosper | \(\frac {x^{5} \left (-2 x^{2}+7 \sqrt {10}\right )}{350 \left (-x^{2}+\sqrt {10}\right )^{\frac {7}{2}}}\) | \(28\) |
pseudoelliptic | \(\frac {x^{5} \left (-2 x^{2}+7 \sqrt {10}\right )}{350 \left (-x^{2}+\sqrt {10}\right )^{\frac {7}{2}}}\) | \(28\) |
meijerg | \(\frac {10^{\frac {3}{4}} x^{5} \left (-\frac {\sqrt {2}\, \sqrt {5}\, x^{2}}{5}+7\right )}{35000 \left (1-\frac {\sqrt {10}\, x^{2}}{10}\right )^{\frac {7}{2}}}\) | \(34\) |
risch | \(\frac {2 x^{7}-7 \sqrt {10}\, x^{5}}{350 \left (x^{2}-\sqrt {10}\right )^{3} \sqrt {-x^{2}+\sqrt {10}}}\) | \(39\) |
trager | \(-\frac {2 \sqrt {10}\, \left (\sqrt {10}\, x^{2}-35\right ) x^{5} \sqrt {-x^{2}+\sqrt {10}}}{35 \left (\sqrt {10}\, x^{2}-10\right )^{4}}\) | \(40\) |
default | \(\frac {x^{3}}{4 \left (-x^{2}+\sqrt {10}\right )^{\frac {7}{2}}}-\frac {3 \sqrt {10}\, \left (\frac {x}{6 \left (-x^{2}+\sqrt {10}\right )^{\frac {7}{2}}}-\frac {\sqrt {10}\, \left (\frac {x \sqrt {10}}{70 \left (-x^{2}+\sqrt {10}\right )^{\frac {7}{2}}}+\frac {3 \sqrt {10}\, \left (\frac {x \sqrt {10}}{50 \left (-x^{2}+\sqrt {10}\right )^{\frac {5}{2}}}+\frac {2 \sqrt {10}\, \left (\frac {x \sqrt {10}}{30 \left (-x^{2}+\sqrt {10}\right )^{\frac {3}{2}}}+\frac {x}{15 \sqrt {-x^{2}+\sqrt {10}}}\right )}{25}\right )}{35}\right )}{6}\right )}{4}\) | \(121\) |
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none
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.38 \[ \int \frac {x^4}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx=-\frac {{\left (2 \, x^{15} - 160 \, x^{11} - 2600 \, x^{7} + \sqrt {10} {\left (x^{13} - 340 \, x^{9} - 700 \, x^{5}\right )}\right )} \sqrt {-x^{2} + \sqrt {10}}}{350 \, {\left (x^{16} - 40 \, x^{12} + 600 \, x^{8} - 4000 \, x^{4} + 10000\right )}} \]
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Timed out. \[ \int \frac {x^4}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (36) = 72\).
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.58 \[ \int \frac {x^4}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx=\frac {x}{175 \, \sqrt {-x^{2} + \sqrt {10}}} + \frac {\sqrt {10} x}{350 \, {\left (-x^{2} + \sqrt {10}\right )}^{\frac {3}{2}}} + \frac {x^{3}}{4 \, {\left (-x^{2} + \sqrt {10}\right )}^{\frac {7}{2}}} + \frac {3 \, x}{140 \, {\left (-x^{2} + \sqrt {10}\right )}^{\frac {5}{2}}} - \frac {3 \, \sqrt {10} x}{28 \, {\left (-x^{2} + \sqrt {10}\right )}^{\frac {7}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (36) = 72\).
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.96 \[ \int \frac {x^4}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx=-\frac {16 \, {\left (7 \, {\left (\frac {x}{\sqrt {-x^{2} + \sqrt {10}} - 10^{\frac {1}{4}}} - \frac {\sqrt {-x^{2} + \sqrt {10}} - 10^{\frac {1}{4}}}{x}\right )}^{2} + 20\right )}}{175 \, {\left (\frac {x}{\sqrt {-x^{2} + \sqrt {10}} - 10^{\frac {1}{4}}} - \frac {\sqrt {-x^{2} + \sqrt {10}} - 10^{\frac {1}{4}}}{x}\right )}^{7}} \]
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Timed out. \[ \int \frac {x^4}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx=\int \frac {x^4}{{\left (\sqrt {10}-x^2\right )}^{9/2}} \,d x \]
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