Optimal. Leaf size=27 \[ \frac{2}{9} \left (x^3+1\right )^{3/2}-\frac{2 \sqrt{x^3+1}}{3} \]
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Rubi [A] time = 0.0102721, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{2}{9} \left (x^3+1\right )^{3/2}-\frac{2 \sqrt{x^3+1}}{3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{1+x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt{1+x}}+\sqrt{1+x}\right ) \, dx,x,x^3\right )\\ &=-\frac{2}{3} \sqrt{1+x^3}+\frac{2}{9} \left (1+x^3\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0049559, size = 18, normalized size = 0.67 \[ \frac{2}{9} \left (x^3-2\right ) \sqrt{x^3+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 26, normalized size = 1. \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ({x}^{3}-2 \right ) }{9}{\frac{1}{\sqrt{{x}^{3}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967665, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{9} \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94551, size = 39, normalized size = 1.44 \begin{align*} \frac{2}{9} \, \sqrt{x^{3} + 1}{\left (x^{3} - 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.363514, size = 26, normalized size = 0.96 \begin{align*} \frac{2 x^{3} \sqrt{x^{3} + 1}}{9} - \frac{4 \sqrt{x^{3} + 1}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11529, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{9} \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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