Optimal. Leaf size=53 \[ \frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} \sqrt {1-x^4} x^3+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {288, 321, 307, 221, 1181, 424} \[ \frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} \sqrt {1-x^4} x^3+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 221
Rule 288
Rule 307
Rule 321
Rule 424
Rule 1181
Rubi steps
\begin {align*} \int \frac {x^{10}}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {x^7}{2 \sqrt {1-x^4}}-\frac {7}{2} \int \frac {x^6}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} x^3 \sqrt {1-x^4}-\frac {21}{10} \int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} x^3 \sqrt {1-x^4}+\frac {21}{10} \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {21}{10} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} x^3 \sqrt {1-x^4}+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {21}{10} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=\frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} x^3 \sqrt {1-x^4}-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 49, normalized size = 0.92 \[ -\frac {x^3 \left (-7 \sqrt {1-x^4} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};x^4\right )+x^4+7\right )}{5 \sqrt {1-x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 1} x^{10}}{x^{8} - 2 \, x^{4} + 1}, x\right ) \]
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 \[ \int \frac {x^{10}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 1.28 \[ \frac {x^{3}}{2 \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{4}+1}\, x^{3}}{5}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (x , i\right )+\EllipticF \left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}} \]
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 \[ \int \frac {x^{10}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^{10}}{{\left (1-x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.63, size = 31, normalized size = 0.58 \[ \frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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