Optimal. Leaf size=61 \[ \frac {15}{14} \sqrt {1-x^4} x+\frac {x^9}{2 \sqrt {1-x^4}}+\frac {9}{14} \sqrt {1-x^4} x^5-\frac {15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {288, 321, 221} \[ \frac {x^9}{2 \sqrt {1-x^4}}+\frac {9}{14} \sqrt {1-x^4} x^5+\frac {15}{14} \sqrt {1-x^4} x-\frac {15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 221
Rule 288
Rule 321
Rubi steps
\begin {align*} \int \frac {x^{12}}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {x^9}{2 \sqrt {1-x^4}}-\frac {9}{2} \int \frac {x^8}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^9}{2 \sqrt {1-x^4}}+\frac {9}{14} x^5 \sqrt {1-x^4}-\frac {45}{14} \int \frac {x^4}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^9}{2 \sqrt {1-x^4}}+\frac {15}{14} x \sqrt {1-x^4}+\frac {9}{14} x^5 \sqrt {1-x^4}-\frac {15}{14} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^9}{2 \sqrt {1-x^4}}+\frac {15}{14} x \sqrt {1-x^4}+\frac {9}{14} x^5 \sqrt {1-x^4}-\frac {15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 54, normalized size = 0.89 \[ -\frac {x \left (15 \sqrt {1-x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )+2 x^8+6 x^4-15\right )}{14 \sqrt {1-x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 1} x^{12}}{x^{8} - 2 \, x^{4} + 1}, x\right ) \]
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 \[ \int \frac {x^{12}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 1.16 \[ \frac {\sqrt {-x^{4}+1}\, x^{5}}{7}+\frac {x}{2 \sqrt {-x^{4}+1}}+\frac {4 \sqrt {-x^{4}+1}\, x}{7}-\frac {15 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{14 \sqrt {-x^{4}+1}} \]
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 \[ \int \frac {x^{12}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^{12}}{{\left (1-x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.74, size = 31, normalized size = 0.51 \[ \frac {x^{13} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {17}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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