Integrand size = 19, antiderivative size = 75 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=-\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {3 \sqrt {a x^{23}} \text {arcsinh}\left (x^{5/2}\right )}{20 x^{23/2}} \]
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Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {15, 327, 335, 281, 221} \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\frac {3 \sqrt {a x^{23}} \text {arcsinh}\left (x^{5/2}\right )}{20 x^{23/2}}-\frac {3 \sqrt {x^5+1} \sqrt {a x^{23}}}{20 x^9}+\frac {\sqrt {x^5+1} \sqrt {a x^{23}}}{10 x^4} \]
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Rule 15
Rule 221
Rule 281
Rule 327
Rule 335
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a x^{23}} \int \frac {x^{23/2}}{\sqrt {1+x^5}} \, dx}{x^{23/2}} \\ & = \frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}-\frac {\left (3 \sqrt {a x^{23}}\right ) \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx}{4 x^{23/2}} \\ & = -\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {\left (3 \sqrt {a x^{23}}\right ) \int \frac {x^{3/2}}{\sqrt {1+x^5}} \, dx}{8 x^{23/2}} \\ & = -\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {\left (3 \sqrt {a x^{23}}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{4 x^{23/2}} \\ & = -\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {\left (3 \sqrt {a x^{23}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{5/2}\right )}{20 x^{23/2}} \\ & = -\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {3 \sqrt {a x^{23}} \sinh ^{-1}\left (x^{5/2}\right )}{20 x^{23/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\frac {\sqrt {a x^{23}} \left (x^{5/2} \sqrt {1+x^5} \left (-3+2 x^5\right )+3 \log \left (x^{5/2}+\sqrt {1+x^5}\right )\right )}{20 x^{23/2}} \]
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Time = 1.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.64
method | result | size |
meijerg | \(\frac {\sqrt {a \,x^{23}}\, \left (-\frac {\sqrt {\pi }\, x^{\frac {5}{2}} \left (-10 x^{5}+15\right ) \sqrt {x^{5}+1}}{20}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (x^{\frac {5}{2}}\right )}{4}\right )}{5 x^{\frac {23}{2}} \sqrt {\pi }}\) | \(48\) |
risch | \(\frac {\left (2 x^{5}-3\right ) \sqrt {x^{5}+1}\, \sqrt {a \,x^{23}}}{20 x^{9}}+\frac {3 \,\operatorname {arcsinh}\left (x^{\frac {5}{2}}\right ) \sqrt {a \,x^{23}}\, \sqrt {a x \left (x^{5}+1\right )}}{20 \sqrt {a}\, x^{12} \sqrt {x^{5}+1}}\) | \(64\) |
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none
Time = 0.35 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\left [\frac {3 \, \sqrt {a} x^{9} \log \left (-\frac {8 \, a x^{19} + 8 \, a x^{14} + a x^{9} + 4 \, \sqrt {a x^{23}} {\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {a}}{x^{9}}\right ) + 4 \, \sqrt {a x^{23}} {\left (2 \, x^{5} - 3\right )} \sqrt {x^{5} + 1}}{80 \, x^{9}}, -\frac {3 \, \sqrt {-a} x^{9} \arctan \left (\frac {\sqrt {a x^{23}} {\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {-a}}{2 \, {\left (a x^{19} + a x^{14}\right )}}\right ) - 2 \, \sqrt {a x^{23}} {\left (2 \, x^{5} - 3\right )} \sqrt {x^{5} + 1}}{40 \, x^{9}}\right ] \]
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\[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\int \frac {\sqrt {a x^{23}}}{\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]
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\[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\int { \frac {\sqrt {a x^{23}}}{\sqrt {x^{5} + 1}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (55) = 110\).
Time = 0.39 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\frac {\sqrt {a^{6} x^{5} + a^{6}} \sqrt {a x} a^{6} x^{2} {\left (\frac {2 \, x^{5}}{a^{8}} - \frac {3}{a^{8}}\right )} \mathrm {sgn}\left (x\right )}{20 \, {\left | a \right |}} - \frac {3 \, {\left (\frac {a^{\frac {5}{2}} \log \left (-\sqrt {a x} a^{\frac {5}{2}} x^{2} + \sqrt {a^{6} x^{5} + a^{6}}\right ) \mathrm {sgn}\left (x\right )}{{\left | a \right |}} - \frac {a^{\frac {5}{2}} \log \left (a^{2} {\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{{\left | a \right |}}\right )} a^{3}}{20 \, {\left | a \right |}^{4}} \]
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Timed out. \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\int \frac {\sqrt {a\,x^{23}}}{\sqrt {x^5+1}} \,d x \]
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