Optimal. Leaf size=35 \[ \frac {x^3}{2 \sqrt {1-x^4}}+\frac {1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {290, 307, 221, 1181, 424} \[ \frac {x^3}{2 \sqrt {1-x^4}}+\frac {1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 221
Rule 290
Rule 307
Rule 424
Rule 1181
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {x^3}{2 \sqrt {1-x^4}}-\frac {1}{2} \int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^3}{2 \sqrt {1-x^4}}+\frac {1}{2} \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {1}{2} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^3}{2 \sqrt {1-x^4}}+\frac {1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{2} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=\frac {x^3}{2 \sqrt {1-x^4}}-\frac {1}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 20, normalized size = 0.57 \[ \frac {1}{3} x^3 \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};x^4\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 1} x^{2}}{x^{8} - 2 \, x^{4} + 1}, x\right ) \]
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 \[ \int \frac {x^{2}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 1.54 \[ \frac {x^{3}}{2 \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (x , i\right )+\EllipticF \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}} \]
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 \[ \int \frac {x^{2}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^2}{{\left (1-x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.86, size = 31, normalized size = 0.89 \[ \frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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