Optimal. Leaf size=53 \[ \frac {2}{3} \sqrt {-1+x^3}+\frac {2}{3} \left (-1+x^3\right )^{3/2}+\frac {2}{5} \left (-1+x^3\right )^{5/2}+\frac {2}{21} \left (-1+x^3\right )^{7/2} \]
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Rubi [A]
time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} \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 \sqrt {x^3-1}}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{11}}{\sqrt {-1+x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {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 = 0.02, size = 30, normalized size = 0.57 \begin {gather*} \frac {2}{105} \sqrt {-1+x^3} \left (16+8 x^3+6 x^6+5 x^9\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 47, normalized size = 0.89
method | result | size |
trager | \(\left (\frac {2}{21} x^{9}+\frac {4}{35} x^{6}+\frac {16}{105} x^{3}+\frac {32}{105}\right ) \sqrt {x^{3}-1}\) | \(26\) |
risch | \(\frac {2 \left (5 x^{9}+6 x^{6}+8 x^{3}+16\right ) \sqrt {x^{3}-1}}{105}\) | \(27\) |
gosper | \(\frac {2 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (5 x^{9}+6 x^{6}+8 x^{3}+16\right )}{105 \sqrt {x^{3}-1}}\) | \(36\) |
default | \(\frac {2 x^{9} \sqrt {x^{3}-1}}{21}+\frac {4 x^{6} \sqrt {x^{3}-1}}{35}+\frac {16 x^{3} \sqrt {x^{3}-1}}{105}+\frac {32 \sqrt {x^{3}-1}}{105}\) | \(47\) |
elliptic | \(\frac {2 x^{9} \sqrt {x^{3}-1}}{21}+\frac {4 x^{6} \sqrt {x^{3}-1}}{35}+\frac {16 x^{3} \sqrt {x^{3}-1}}{105}+\frac {32 \sqrt {x^{3}-1}}{105}\) | \(47\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (40 x^{9}+48 x^{6}+64 x^{3}+128\right ) \sqrt {-x^{3}+1}}{140}\right )}{3 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{3}-1\right )}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 37, normalized size = 0.70 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 26, normalized size = 0.49 \begin {gather*} \frac {2}{105} \, {\left (5 \, x^{9} + 6 \, x^{6} + 8 \, x^{3} + 16\right )} \sqrt {x^{3} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 56, normalized size = 1.06 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.46, size = 37, normalized size = 0.70 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 46, normalized size = 0.87 \begin {gather*} \frac {32\,\sqrt {x^3-1}}{105}+\frac {16\,x^3\,\sqrt {x^3-1}}{105}+\frac {4\,x^6\,\sqrt {x^3-1}}{35}+\frac {2\,x^9\,\sqrt {x^3-1}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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