Optimal. Leaf size=10 \[ \frac{2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]
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Rubi [A] time = 0.0067993, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {329, 275, 215} \[ \frac{2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]
Antiderivative was successfully verified.
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Rule 329
Rule 275
Rule 215
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*}
Mathematica [A] time = 0.0025837, size = 10, normalized size = 1. \[ \frac{2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 7, normalized size = 0.7 \begin{align*}{\frac{2}{5}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.954905, size = 45, normalized size = 4.5 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.18181, size = 62, normalized size = 6.2 \begin{align*} \frac{1}{5} \, \log \left (2 \, x^{5} + 2 \, \sqrt{x^{5} + 1} x^{\frac{5}{2}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.46149, size = 8, normalized size = 0.8 \begin{align*} \frac{2 \operatorname{asinh}{\left (x^{\frac{5}{2}} \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22061, size = 34, normalized size = 3.4 \begin{align*} \frac{1}{5} \, \log \left (\sqrt{\frac{1}{x^{5}} + 1} + 1\right ) - \frac{1}{5} \, \log \left (\sqrt{\frac{1}{x^{5}} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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