Optimal. Leaf size=29 \[ \frac {1}{6} \left (16-x^4\right )^{3/2}-8 \sqrt {16-x^4} \]
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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 = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {1}{6} \left (16-x^4\right )^{3/2}-8 \sqrt {16-x^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^7}{\sqrt {16-x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{\sqrt {16-x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {16}{\sqrt {16-x}}-\sqrt {16-x}\right ) \, dx,x,x^4\right )\\ &=-8 \sqrt {16-x^4}+\frac {1}{6} \left (16-x^4\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.69 \[ -\frac {1}{6} \sqrt {16-x^4} \left (x^4+32\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 16, normalized size = 0.55 \[ -\frac {1}{6} \, {\left (x^{4} + 32\right )} \sqrt {-x^{4} + 16} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 23, normalized size = 0.79 \[ \frac {1}{6} \, {\left (-x^{4} + 16\right )}^{\frac {3}{2}} - 8 \, \sqrt {-x^{4} + 16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 0.97 \[ \frac {\left (x -2\right ) \left (x +2\right ) \left (x^{2}+4\right ) \left (x^{4}+32\right )}{6 \sqrt {-x^{4}+16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 23, normalized size = 0.79 \[ \frac {1}{6} \, {\left (-x^{4} + 16\right )}^{\frac {3}{2}} - 8 \, \sqrt {-x^{4} + 16} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 16, normalized size = 0.55 \[ -\frac {\sqrt {16-x^4}\,\left (x^4+32\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 26, normalized size = 0.90 \[ - \frac {x^{4} \sqrt {16 - x^{4}}}{6} - \frac {16 \sqrt {16 - x^{4}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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