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 = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 43
Rule 266
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.54, size = 23, normalized size = 0.50 \[ -\frac {2}{45} \, {\left (3 \, x^{6} + 4 \, x^{3} + 8\right )} \sqrt {-x^{3} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 41, normalized size = 0.89 \[ -\frac {2}{15} \, {\left (x^{3} - 1\right )}^{2} \sqrt {-x^{3} + 1} + \frac {4}{9} \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {-x^{3} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 33, normalized size = 0.72 \[ \frac {2 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (3 x^{6}+4 x^{3}+8\right )}{45 \sqrt {-x^{3}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 34, normalized size = 0.74 \[ -\frac {2}{15} \, {\left (-x^{3} + 1\right )}^{\frac {5}{2}} + \frac {4}{9} \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {-x^{3} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 23, normalized size = 0.50 \[ -\frac {2\,\sqrt {1-x^3}\,\left (3\,x^6+4\,x^3+8\right )}{45} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.78, size = 42, normalized size = 0.91 \[ - \frac {2 x^{6} \sqrt {1 - x^{3}}}{15} - \frac {8 x^{3} \sqrt {1 - x^{3}}}{45} - \frac {16 \sqrt {1 - x^{3}}}{45} \]
Verification of antiderivative is not currently implemented for this CAS.
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