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 = 0.02, 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} \[ -\frac {2}{15} \left (-x^3-1\right )^{5/2}-\frac {4}{9} \left (-x^3-1\right )^{3/2}-\frac {2}{3} \sqrt {-x^3-1} \]
Antiderivative was successfully verified.
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Rule 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 \[ -\frac {2}{45} \sqrt {-x^3-1} \left (3 x^6-4 x^3+8\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, 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 = 0.00, size = 35, normalized size = 0.76 \[ \frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \left (3 x^{6}-4 x^{3}+8\right )}{45 \sqrt {-x^{3}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, 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 \[ -\frac {2\,\sqrt {-x^3-1}\,\left (3\,x^6-4\,x^3+8\right )}{45} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 46, normalized size = 1.00 \[ - \frac {2 x^{6} \sqrt {- x^{3} - 1}}{15} + \frac {8 x^{3} \sqrt {- x^{3} - 1}}{45} - \frac {16 \sqrt {- x^{3} - 1}}{45} \]
Verification of antiderivative is not currently implemented for this CAS.
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