Optimal. Leaf size=61 \[ \frac{2}{21} \left (-x^3-1\right )^{7/2}+\frac{2}{5} \left (-x^3-1\right )^{5/2}+\frac{2}{3} \left (-x^3-1\right )^{3/2}+\frac{2}{3} \sqrt{-x^3-1} \]
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Rubi [A] time = 0.0248423, antiderivative size = 61, 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} \[ \frac{2}{21} \left (-x^3-1\right )^{7/2}+\frac{2}{5} \left (-x^3-1\right )^{5/2}+\frac{2}{3} \left (-x^3-1\right )^{3/2}+\frac{2}{3} \sqrt{-x^3-1} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\sqrt{-1-x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{-1-x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt{-1-x}}-3 \sqrt{-1-x}-3 (-1-x)^{3/2}-(-1-x)^{5/2}\right ) \, dx,x,x^3\right )\\ &=\frac{2}{3} \sqrt{-1-x^3}+\frac{2}{3} \left (-1-x^3\right )^{3/2}+\frac{2}{5} \left (-1-x^3\right )^{5/2}+\frac{2}{21} \left (-1-x^3\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0106054, size = 32, normalized size = 0.52 \[ -\frac{2}{105} \sqrt{-x^3-1} \left (5 x^9-6 x^6+8 x^3-16\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 40, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ( 5\,{x}^{9}-6\,{x}^{6}+8\,{x}^{3}-16 \right ) }{105}{\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 = 1.19021, size = 61, normalized size = 1. \begin{align*} \frac{2}{21} \,{\left (-x^{3} - 1\right )}^{\frac{7}{2}} + \frac{2}{5} \,{\left (-x^{3} - 1\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\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 = 1.7713, size = 70, normalized size = 1.15 \begin{align*} -\frac{2}{105} \,{\left (5 \, x^{9} - 6 \, x^{6} + 8 \, x^{3} - 16\right )} \sqrt{-x^{3} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.77158, size = 63, normalized size = 1.03 \begin{align*} - \frac{2 x^{9} \sqrt{- x^{3} - 1}}{21} + \frac{4 x^{6} \sqrt{- x^{3} - 1}}{35} - \frac{16 x^{3} \sqrt{- x^{3} - 1}}{105} + \frac{32 \sqrt{- x^{3} - 1}}{105} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10949, size = 80, normalized size = 1.31 \begin{align*} -\frac{2}{21} \,{\left (x^{3} + 1\right )}^{3} \sqrt{-x^{3} - 1} + \frac{2}{5} \,{\left (x^{3} + 1\right )}^{2} \sqrt{-x^{3} - 1} + \frac{2}{3} \,{\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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