Optimal. Leaf size=46 \[ -\frac{2}{15} \left (1-x^3\right )^{5/2}+\frac{4}{9} \left (1-x^3\right )^{3/2}-\frac{2 \sqrt{1-x^3}}{3} \]
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Rubi [A] time = 0.0186933, 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} \[ -\frac{2}{15} \left (1-x^3\right )^{5/2}+\frac{4}{9} \left (1-x^3\right )^{3/2}-\frac{2 \sqrt{1-x^3}}{3} \]
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 = 0.0086716, size = 27, normalized size = 0.59 \[ -\frac{2}{45} \sqrt{1-x^3} \left (3 x^6+4 x^3+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 33, normalized size = 0.7 \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 = 0.999418, 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 = 1.4613, 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 = 0.897993, size = 42, normalized size = 0.91 \begin{align*} - \frac{2 x^{6} \sqrt{1 - x^{3}}}{15} - \frac{8 x^{3} \sqrt{1 - x^{3}}}{45} - \frac{16 \sqrt{1 - x^{3}}}{45} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1423, 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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