Optimal. Leaf size=29 \[ 4 \sin ^{-1}\left (\frac {x^2}{4}\right )-\frac {1}{4} x^2 \sqrt {16-x^4} \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {275, 321, 216} \[ 4 \sin ^{-1}\left (\frac {x^2}{4}\right )-\frac {1}{4} x^2 \sqrt {16-x^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 275
Rule 321
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {16-x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {16-x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{4} x^2 \sqrt {16-x^4}+4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {16-x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{4} x^2 \sqrt {16-x^4}+4 \sin ^{-1}\left (\frac {x^2}{4}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 1.00 \[ 4 \sin ^{-1}\left (\frac {x^2}{4}\right )-\frac {1}{4} x^2 \sqrt {16-x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 33, normalized size = 1.14 \[ -\frac {1}{4} \, \sqrt {-x^{4} + 16} x^{2} - 8 \, \arctan \left (\frac {\sqrt {-x^{4} + 16} - 4}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 23, normalized size = 0.79 \[ -\frac {1}{4} \, \sqrt {-x^{4} + 16} x^{2} + 4 \, \arcsin \left (\frac {1}{4} \, x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 24, normalized size = 0.83 \[ -\frac {\sqrt {-x^{4}+16}\, x^{2}}{4}+4 \arcsin \left (\frac {x^{2}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 44, normalized size = 1.52 \[ \frac {4 \, \sqrt {-x^{4} + 16}}{x^{2} {\left (\frac {x^{4} - 16}{x^{4}} - 1\right )}} - 4 \, \arctan \left (\frac {\sqrt {-x^{4} + 16}}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^5}{\sqrt {16-x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.89, size = 80, normalized size = 2.76 \[ \begin {cases} - \frac {i x^{6}}{4 \sqrt {x^{4} - 16}} + \frac {4 i x^{2}}{\sqrt {x^{4} - 16}} - 4 i \operatorname {acosh}{\left (\frac {x^{2}}{4} \right )} & \text {for}\: \frac {\left |{x^{4}}\right |}{16} > 1 \\\frac {x^{6}}{4 \sqrt {16 - x^{4}}} - \frac {4 x^{2}}{\sqrt {16 - x^{4}}} + 4 \operatorname {asin}{\left (\frac {x^{2}}{4} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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