Optimal. Leaf size=10 \[ \frac {2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {329, 275, 215} \[ \frac {2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 275
Rule 329
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {1+x^5}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{5/2}\right )\\ &=\frac {2}{5} \sinh ^{-1}\left (x^{5/2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 10, normalized size = 1.00 \[ \frac {2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 22, normalized size = 2.20 \[ \frac {1}{5} \, \log \left (2 \, x^{5} + 2 \, \sqrt {x^{5} + 1} x^{\frac {5}{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 16, normalized size = 1.60 \[ -\frac {2}{5} \, \log \left (-x^{\frac {5}{2}} + \sqrt {x^{5} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 7, normalized size = 0.70 \[ \frac {2 \arcsinh \left (x^{\frac {5}{2}}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.93, size = 33, normalized size = 3.30 \[ \frac {1}{5} \, \log \left (\frac {\sqrt {x^{5} + 1}}{x^{\frac {5}{2}}} + 1\right ) - \frac {1}{5} \, \log \left (\frac {\sqrt {x^{5} + 1}}{x^{\frac {5}{2}}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.10 \[ \int \frac {x^{3/2}}{\sqrt {x^5+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.38, size = 8, normalized size = 0.80 \[ \frac {2 \operatorname {asinh}{\left (x^{\frac {5}{2}} \right )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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