Optimal. Leaf size=61 \[ \frac {2}{21} \left (-x^3-1\right )^{7/2}+\frac {2}{5} \left (-x^3-1\right )^{5/2}+\frac {2}{3} \left (-x^3-1\right )^{3/2}+\frac {2}{3} \sqrt {-x^3-1} \]
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Rubi [A] time = 0.02, antiderivative size = 61, 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} \[ \frac {2}{21} \left (-x^3-1\right )^{7/2}+\frac {2}{5} \left (-x^3-1\right )^{5/2}+\frac {2}{3} \left (-x^3-1\right )^{3/2}+\frac {2}{3} \sqrt {-x^3-1} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{11}}{\sqrt {-1-x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {-1-x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {-1-x}}-3 \sqrt {-1-x}-3 (-1-x)^{3/2}-(-1-x)^{5/2}\right ) \, dx,x,x^3\right )\\ &=\frac {2}{3} \sqrt {-1-x^3}+\frac {2}{3} \left (-1-x^3\right )^{3/2}+\frac {2}{5} \left (-1-x^3\right )^{5/2}+\frac {2}{21} \left (-1-x^3\right )^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.52 \[ -\frac {2}{105} \sqrt {-x^3-1} \left (5 x^9-6 x^6+8 x^3-16\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 28, normalized size = 0.46 \[ -\frac {2}{105} \, {\left (5 \, x^{9} - 6 \, x^{6} + 8 \, x^{3} - 16\right )} \sqrt {-x^{3} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 59, normalized size = 0.97 \[ -\frac {2}{21} \, {\left (x^{3} + 1\right )}^{3} \sqrt {-x^{3} - 1} + \frac {2}{5} \, {\left (x^{3} + 1\right )}^{2} \sqrt {-x^{3} - 1} + \frac {2}{3} \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {-x^{3} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.66 \[ \frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \left (5 x^{9}-6 x^{6}+8 x^{3}-16\right )}{105 \sqrt {-x^{3}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 45, normalized size = 0.74 \[ \frac {2}{21} \, {\left (-x^{3} - 1\right )}^{\frac {7}{2}} + \frac {2}{5} \, {\left (-x^{3} - 1\right )}^{\frac {5}{2}} + \frac {2}{3} \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {-x^{3} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 54, normalized size = 0.89 \[ \frac {4\,x^6\,\sqrt {-x^3-1}}{35}-\frac {16\,x^3\,\sqrt {-x^3-1}}{105}-\frac {2\,x^9\,\sqrt {-x^3-1}}{21}+\frac {32\,\sqrt {-x^3-1}}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.90, size = 63, normalized size = 1.03 \[ - \frac {2 x^{9} \sqrt {- x^{3} - 1}}{21} + \frac {4 x^{6} \sqrt {- x^{3} - 1}}{35} - \frac {16 x^{3} \sqrt {- x^{3} - 1}}{105} + \frac {32 \sqrt {- x^{3} - 1}}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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