Optimal. Leaf size=31 \[ \frac {2}{9} \left (-x^3-1\right )^{3/2}+\frac {2}{3} \sqrt {-x^3-1} \]
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Rubi [A] time = 0.01, antiderivative size = 31, 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} \[ \frac {2}{9} \left (-x^3-1\right )^{3/2}+\frac {2}{3} \sqrt {-x^3-1} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {-1-x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1-x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {-1-x}}-\sqrt {-1-x}\right ) \, dx,x,x^3\right )\\ &=\frac {2}{3} \sqrt {-1-x^3}+\frac {2}{9} \left (-1-x^3\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.65 \[ -\frac {2}{9} \sqrt {-x^3-1} \left (x^3-2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 16, normalized size = 0.52 \[ -\frac {2}{9} \, {\left (x^{3} - 2\right )} \sqrt {-x^{3} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 23, normalized size = 0.74 \[ \frac {2}{9} \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {-x^{3} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 0.90 \[ \frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \left (x^{3}-2\right )}{9 \sqrt {-x^{3}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 23, normalized size = 0.74 \[ \frac {2}{9} \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {-x^{3} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 16, normalized size = 0.52 \[ -\frac {2\,\sqrt {-x^3-1}\,\left (x^3-2\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 29, normalized size = 0.94 \[ - \frac {2 x^{3} \sqrt {- x^{3} - 1}}{9} + \frac {4 \sqrt {- x^{3} - 1}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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