Optimal. Leaf size=46 \[ -\frac {2}{3} \sqrt {-1-x^3}-\frac {4}{9} \left (-1-x^3\right )^{3/2}-\frac {2}{15} \left (-1-x^3\right )^{5/2} \]
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Rubi [A]
time = 0.01, 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 = {272, 45}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {-1-x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\sqrt {-1-x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {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 = 0.02, size = 27, normalized size = 0.59 \begin {gather*} -\frac {2}{45} \sqrt {-1-x^3} \left (8-4 x^3+3 x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 41, normalized size = 0.89
method | result | size |
trager | \(\left (-\frac {2}{15} x^{6}+\frac {8}{45} x^{3}-\frac {16}{45}\right ) \sqrt {-x^{3}-1}\) | \(23\) |
risch | \(\frac {2 \left (3 x^{6}-4 x^{3}+8\right ) \left (x^{3}+1\right )}{45 \sqrt {-x^{3}-1}}\) | \(29\) |
gosper | \(\frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \left (3 x^{6}-4 x^{3}+8\right )}{45 \sqrt {-x^{3}-1}}\) | \(35\) |
meijerg | \(-\frac {i \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 x^{6}-8 x^{3}+16\right ) \sqrt {x^{3}+1}}{15}\right )}{3 \sqrt {\pi }}\) | \(37\) |
default | \(-\frac {2 x^{6} \sqrt {-x^{3}-1}}{15}+\frac {8 x^{3} \sqrt {-x^{3}-1}}{45}-\frac {16 \sqrt {-x^{3}-1}}{45}\) | \(41\) |
elliptic | \(-\frac {2 x^{6} \sqrt {-x^{3}-1}}{15}+\frac {8 x^{3} \sqrt {-x^{3}-1}}{45}-\frac {16 \sqrt {-x^{3}-1}}{45}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 34, normalized size = 0.74 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 23, normalized size = 0.50 \begin {gather*} -\frac {2}{45} \, {\left (3 \, x^{6} - 4 \, x^{3} + 8\right )} \sqrt {-x^{3} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 46, normalized size = 1.00 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.40, size = 41, normalized size = 0.89 \begin {gather*} -\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 {gather*}
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 \begin {gather*} -\frac {2\,\sqrt {-x^3-1}\,\left (3\,x^6-4\,x^3+8\right )}{45} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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