Optimal. Leaf size=25 \[ \frac {2}{77} \left (x^2-1\right )^{3/4} \left (7 x^4-3 x^2-4\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.08, 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 = {266, 43} \begin {gather*} \frac {2}{11} \left (x^2-1\right )^{11/4}+\frac {2}{7} \left (x^2-1\right )^{7/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^3 \left (-1+x^2\right )^{3/4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (-1+x)^{3/4} x \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left ((-1+x)^{3/4}+(-1+x)^{7/4}\right ) \, dx,x,x^2\right )\\ &=\frac {2}{7} \left (-1+x^2\right )^{7/4}+\frac {2}{11} \left (-1+x^2\right )^{11/4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {2}{77} \left (x^2-1\right )^{7/4} \left (7 x^2+4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 20, normalized size = 0.80 \begin {gather*} \frac {2}{77} \left (-1+x^2\right )^{7/4} \left (4+7 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 21, normalized size = 0.84 \begin {gather*} \frac {2}{77} \, {\left (7 \, x^{4} - 3 \, x^{2} - 4\right )} {\left (x^{2} - 1\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 19, normalized size = 0.76 \begin {gather*} \frac {2}{11} \, {\left (x^{2} - 1\right )}^{\frac {11}{4}} + \frac {2}{7} \, {\left (x^{2} - 1\right )}^{\frac {7}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 21, normalized size = 0.84
method | result | size |
trager | \(\left (\frac {2}{11} x^{4}-\frac {6}{77} x^{2}-\frac {8}{77}\right ) \left (x^{2}-1\right )^{\frac {3}{4}}\) | \(21\) |
risch | \(\frac {2 \left (x^{2}-1\right )^{\frac {3}{4}} \left (7 x^{4}-3 x^{2}-4\right )}{77}\) | \(22\) |
gosper | \(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (7 x^{2}+4\right ) \left (x^{2}-1\right )^{\frac {3}{4}}}{77}\) | \(23\) |
meijerg | \(\frac {\mathrm {signum}\left (x^{2}-1\right )^{\frac {3}{4}} \hypergeom \left (\left [-\frac {3}{4}, 2\right ], \relax [3], x^{2}\right ) x^{4}}{4 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {3}{4}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 19, normalized size = 0.76 \begin {gather*} \frac {2}{11} \, {\left (x^{2} - 1\right )}^{\frac {11}{4}} + \frac {2}{7} \, {\left (x^{2} - 1\right )}^{\frac {7}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 21, normalized size = 0.84 \begin {gather*} -{\left (x^2-1\right )}^{3/4}\,\left (-\frac {2\,x^4}{11}+\frac {6\,x^2}{77}+\frac {8}{77}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.76, size = 41, normalized size = 1.64 \begin {gather*} \frac {2 x^{4} \left (x^{2} - 1\right )^{\frac {3}{4}}}{11} - \frac {6 x^{2} \left (x^{2} - 1\right )^{\frac {3}{4}}}{77} - \frac {8 \left (x^{2} - 1\right )^{\frac {3}{4}}}{77} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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