Integrand size = 15, antiderivative size = 29 \[ \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx=\frac {1}{5} x^{5/2} \sqrt {1+x^5}-\frac {1}{5} \text {arcsinh}\left (x^{5/2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {327, 335, 281, 221} \[ \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx=\frac {1}{5} x^{5/2} \sqrt {x^5+1}-\frac {1}{5} \text {arcsinh}\left (x^{5/2}\right ) \]
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Rule 221
Rule 281
Rule 327
Rule 335
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^{5/2} \sqrt {1+x^5}-\frac {1}{2} \int \frac {x^{3/2}}{\sqrt {1+x^5}} \, dx \\ & = \frac {1}{5} x^{5/2} \sqrt {1+x^5}-\text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{5} x^{5/2} \sqrt {1+x^5}-\frac {1}{5} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{5/2}\right ) \\ & = \frac {1}{5} x^{5/2} \sqrt {1+x^5}-\frac {1}{5} \sinh ^{-1}\left (x^{5/2}\right ) \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx=\frac {1}{5} x^{5/2} \sqrt {1+x^5}-\frac {1}{5} \log \left (x^{5/2}+\sqrt {1+x^5}\right ) \]
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Time = 4.59 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
method | result | size |
meijerg | \(\frac {\sqrt {\pi }\, x^{\frac {5}{2}} \sqrt {x^{5}+1}-\sqrt {\pi }\, \operatorname {arcsinh}\left (x^{\frac {5}{2}}\right )}{5 \sqrt {\pi }}\) | \(30\) |
risch | \(\frac {x^{\frac {5}{2}} \sqrt {x^{5}+1}}{5}-\frac {\operatorname {arcsinh}\left (x^{\frac {5}{2}}\right ) \sqrt {x \left (x^{5}+1\right )}}{5 \sqrt {x}\, \sqrt {x^{5}+1}}\) | \(39\) |
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Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx=\frac {1}{5} \, \sqrt {x^{5} + 1} x^{\frac {5}{2}} + \frac {1}{10} \, \log \left (-2 \, x^{5} + 2 \, \sqrt {x^{5} + 1} x^{\frac {5}{2}} - 1\right ) \]
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Time = 101.72 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx=\frac {x^{\frac {5}{2}} \sqrt {x^{5} + 1}}{5} - \frac {\operatorname {asinh}{\left (x^{\frac {5}{2}} \right )}}{5} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (19) = 38\).
Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00 \[ \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx=\frac {\sqrt {x^{5} + 1}}{5 \, x^{\frac {5}{2}} {\left (\frac {x^{5} + 1}{x^{5}} - 1\right )}} - \frac {1}{10} \, \log \left (\frac {\sqrt {x^{5} + 1}}{x^{\frac {5}{2}}} + 1\right ) + \frac {1}{10} \, \log \left (\frac {\sqrt {x^{5} + 1}}{x^{\frac {5}{2}}} - 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx=\frac {1}{5} \, \sqrt {x^{5} + 1} x^{\frac {5}{2}} + \frac {1}{5} \, \log \left (-x^{\frac {5}{2}} + \sqrt {x^{5} + 1}\right ) \]
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Timed out. \[ \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx=\int \frac {x^{13/2}}{\sqrt {x^5+1}} \,d x \]
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