Optimal. Leaf size=46 \[ -\frac{2}{15} \left (-x^3-1\right )^{5/2}-\frac{4}{9} \left (-x^3-1\right )^{3/2}-\frac{2}{3} \sqrt{-x^3-1} \]
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Rubi [A] time = 0.0185338, antiderivative size = 46, 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} \[ -\frac{2}{15} \left (-x^3-1\right )^{5/2}-\frac{4}{9} \left (-x^3-1\right )^{3/2}-\frac{2}{3} \sqrt{-x^3-1} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^8}{\sqrt{-1-x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-1-x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{-1-x}}+2 \sqrt{-1-x}+(-1-x)^{3/2}\right ) \, dx,x,x^3\right )\\ &=-\frac{2}{3} \sqrt{-1-x^3}-\frac{4}{9} \left (-1-x^3\right )^{3/2}-\frac{2}{15} \left (-1-x^3\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0077563, size = 27, normalized size = 0.59 \[ -\frac{2}{45} \sqrt{-x^3-1} \left (3 x^6-4 x^3+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 35, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ( 3\,{x}^{6}-4\,{x}^{3}+8 \right ) }{45}{\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 = 1.02993, size = 46, normalized size = 1. \begin{align*} -\frac{2}{15} \,{\left (-x^{3} - 1\right )}^{\frac{5}{2}} - \frac{4}{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 = 1.63207, size = 57, normalized size = 1.24 \begin{align*} -\frac{2}{45} \,{\left (3 \, x^{6} - 4 \, x^{3} + 8\right )} \sqrt{-x^{3} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.824093, size = 46, normalized size = 1. \begin{align*} - \frac{2 x^{6} \sqrt{- x^{3} - 1}}{15} + \frac{8 x^{3} \sqrt{- x^{3} - 1}}{45} - \frac{16 \sqrt{- x^{3} - 1}}{45} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10191, size = 55, normalized size = 1.2 \begin{align*} -\frac{2}{15} \,{\left (x^{3} + 1\right )}^{2} \sqrt{-x^{3} - 1} - \frac{4}{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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