Integrand size = 19, antiderivative size = 24 \[ \int \frac {\sqrt {a x^3}}{\sqrt {1+x^5}} \, dx=\frac {2 \sqrt {a x^3} \text {arcsinh}\left (x^{5/2}\right )}{5 x^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {15, 335, 281, 221} \[ \int \frac {\sqrt {a x^3}}{\sqrt {1+x^5}} \, dx=\frac {2 \sqrt {a x^3} \text {arcsinh}\left (x^{5/2}\right )}{5 x^{3/2}} \]
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Rule 15
Rule 221
Rule 281
Rule 335
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a x^3} \int \frac {x^{3/2}}{\sqrt {1+x^5}} \, dx}{x^{3/2}} \\ & = \frac {\left (2 \sqrt {a x^3}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{x^{3/2}} \\ & = \frac {\left (2 \sqrt {a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{5/2}\right )}{5 x^{3/2}} \\ & = \frac {2 \sqrt {a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {a x^3}}{\sqrt {1+x^5}} \, dx=\frac {2 \sqrt {a x^3} \log \left (x^{5/2}+\sqrt {1+x^5}\right )}{5 x^{3/2}} \]
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Time = 0.96 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71
method | result | size |
meijerg | \(\frac {2 \,\operatorname {arcsinh}\left (x^{\frac {5}{2}}\right ) \sqrt {a \,x^{3}}}{5 x^{\frac {3}{2}}}\) | \(17\) |
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (16) = 32\).
Time = 0.36 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.08 \[ \int \frac {\sqrt {a x^3}}{\sqrt {1+x^5}} \, dx=\left [\frac {1}{10} \, \sqrt {a} \log \left (-8 \, a x^{10} - 8 \, a x^{5} - 4 \, {\left (2 \, x^{6} + x\right )} \sqrt {x^{5} + 1} \sqrt {a x^{3}} \sqrt {a} - a\right ), -\frac {1}{5} \, \sqrt {-a} \arctan \left (\frac {{\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {a x^{3}} \sqrt {-a}}{2 \, {\left (a x^{9} + a x^{4}\right )}}\right )\right ] \]
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\[ \int \frac {\sqrt {a x^3}}{\sqrt {1+x^5}} \, dx=\int \frac {\sqrt {a x^{3}}}{\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]
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\[ \int \frac {\sqrt {a x^3}}{\sqrt {1+x^5}} \, dx=\int { \frac {\sqrt {a x^{3}}}{\sqrt {x^{5} + 1}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (16) = 32\).
Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.42 \[ \int \frac {\sqrt {a x^3}}{\sqrt {1+x^5}} \, dx=-\frac {2 \, a^{\frac {3}{2}} \log \left (-\sqrt {a x} a^{\frac {5}{2}} x^{2} + \sqrt {a^{6} x^{5} + a^{6}}\right ) \mathrm {sgn}\left (x\right )}{5 \, {\left | a \right |}} + \frac {2 \, a^{\frac {3}{2}} \log \left (a^{2} {\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{5 \, {\left | a \right |}} \]
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Timed out. \[ \int \frac {\sqrt {a x^3}}{\sqrt {1+x^5}} \, dx=\int \frac {\sqrt {a\,x^3}}{\sqrt {x^5+1}} \,d x \]
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