Optimal. Leaf size=41 \[ \frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {2}{7} x \sqrt {1-x^4}+\frac {4}{7} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {195, 221} \[ \frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {2}{7} x \sqrt {1-x^4}+\frac {4}{7} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 221
Rubi steps
\begin {align*} \int \left (1-x^4\right )^{3/2} \, dx &=\frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {6}{7} \int \sqrt {1-x^4} \, dx\\ &=\frac {2}{7} x \sqrt {1-x^4}+\frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {4}{7} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=\frac {2}{7} x \sqrt {1-x^4}+\frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {4}{7} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 15, normalized size = 0.37 \[ x \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};x^4\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-x^{4} + 1\right )}^{\frac {3}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-x^{4} + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 1.44 \[ -\frac {\sqrt {-x^{4}+1}\, x^{5}}{7}+\frac {3 \sqrt {-x^{4}+1}\, x}{7}+\frac {4 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{7 \sqrt {-x^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-x^{4} + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 10, normalized size = 0.24 \[ x\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{4};\ \frac {5}{4};\ x^4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.31, size = 31, normalized size = 0.76 \[ \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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