Integrand size = 13, antiderivative size = 28 \[ \int \frac {\left (5+x^2\right )^2}{x^{13/3}} \, dx=-\frac {15}{2 x^{10/3}}-\frac {15}{2 x^{4/3}}+\frac {3 x^{2/3}}{2} \]
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Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \[ \int \frac {\left (5+x^2\right )^2}{x^{13/3}} \, dx=\frac {3 x^{2/3}}{2}-\frac {15}{2 x^{4/3}}-\frac {15}{2 x^{10/3}} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {25}{x^{13/3}}+\frac {10}{x^{7/3}}+\frac {1}{\sqrt [3]{x}}\right ) \, dx \\ & = -\frac {15}{2 x^{10/3}}-\frac {15}{2 x^{4/3}}+\frac {3 x^{2/3}}{2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {\left (5+x^2\right )^2}{x^{13/3}} \, dx=\frac {3 \left (-5-5 x^2+x^4\right )}{2 x^{10/3}} \]
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Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(\frac {\frac {3}{2} x^{4}-\frac {15}{2}-\frac {15}{2} x^{2}}{x^{\frac {10}{3}}}\) | \(16\) |
trager | \(\frac {\frac {3}{2} x^{4}-\frac {15}{2}-\frac {15}{2} x^{2}}{x^{\frac {10}{3}}}\) | \(16\) |
risch | \(\frac {\frac {3}{2} x^{4}-\frac {15}{2}-\frac {15}{2} x^{2}}{x^{\frac {10}{3}}}\) | \(16\) |
derivativedivides | \(-\frac {15}{2 x^{\frac {10}{3}}}-\frac {15}{2 x^{\frac {4}{3}}}+\frac {3 x^{\frac {2}{3}}}{2}\) | \(17\) |
default | \(-\frac {15}{2 x^{\frac {10}{3}}}-\frac {15}{2 x^{\frac {4}{3}}}+\frac {3 x^{\frac {2}{3}}}{2}\) | \(17\) |
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Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.54 \[ \int \frac {\left (5+x^2\right )^2}{x^{13/3}} \, dx=\frac {3 \, {\left (x^{4} - 5 \, x^{2} - 5\right )}}{2 \, x^{\frac {10}{3}}} \]
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Time = 0.81 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (5+x^2\right )^2}{x^{13/3}} \, dx=\frac {3 x^{\frac {2}{3}}}{2} - \frac {15}{2 x^{\frac {4}{3}}} - \frac {15}{2 x^{\frac {10}{3}}} \]
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Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.57 \[ \int \frac {\left (5+x^2\right )^2}{x^{13/3}} \, dx=\frac {3}{2} \, x^{\frac {2}{3}} - \frac {15 \, {\left (x^{2} + 1\right )}}{2 \, x^{\frac {10}{3}}} \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.57 \[ \int \frac {\left (5+x^2\right )^2}{x^{13/3}} \, dx=\frac {3}{2} \, x^{\frac {2}{3}} - \frac {15 \, {\left (x^{2} + 1\right )}}{2 \, x^{\frac {10}{3}}} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {\left (5+x^2\right )^2}{x^{13/3}} \, dx=-\frac {-3\,x^4+15\,x^2+15}{2\,x^{10/3}} \]
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