Optimal. Leaf size=31 \[ \frac{2}{9} \left (-x^3-1\right )^{3/2}+\frac{2}{3} \sqrt{-x^3-1} \]
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Rubi [A] time = 0.0144815, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{2}{9} \left (-x^3-1\right )^{3/2}+\frac{2}{3} \sqrt{-x^3-1} \]
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 = 0.005948, size = 20, normalized size = 0.65 \[ -\frac{2}{9} \sqrt{-x^3-1} \left (x^3-2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 28, normalized size = 0.9 \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 = 1.00286, size = 31, normalized size = 1. \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 = 1.68194, size = 42, normalized size = 1.35 \begin{align*} -\frac{2}{9} \,{\left (x^{3} - 2\right )} \sqrt{-x^{3} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.357661, size = 29, normalized size = 0.94 \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 = 1.09812, size = 31, normalized size = 1. \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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