Optimal. Leaf size=54 \[ \frac {\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}-\frac {\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}} \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {307, 221, 1199, 424} \[ \frac {\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}-\frac {\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 307
Rule 424
Rule 1199
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {3-b x^4}} \, dx &=-\frac {\sqrt {3} \int \frac {1}{\sqrt {3-b x^4}} \, dx}{\sqrt {b}}+\frac {\sqrt {3} \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {3}}}{\sqrt {3-b x^4}} \, dx}{\sqrt {b}}\\ &=-\frac {\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}+\frac {\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {3}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {3}}}} \, dx}{\sqrt {b}}\\ &=\frac {\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}-\frac {\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.56 \[ \frac {x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {b x^4}{3}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{4} + 3} x^{2}}{b x^{4} - 3}, x\right ) \]
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 \[ \int \frac {x^{2}}{\sqrt {-b x^{4} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 94, normalized size = 1.74 \[ -\frac {\sqrt {-3 \sqrt {3}\, \sqrt {b}\, x^{2}+9}\, \sqrt {3 \sqrt {3}\, \sqrt {b}\, x^{2}+9}\, \left (-\EllipticE \left (\frac {\sqrt {3}\, \sqrt {\sqrt {3}\, \sqrt {b}}\, x}{3}, i\right )+\EllipticF \left (\frac {\sqrt {3}\, \sqrt {\sqrt {3}\, \sqrt {b}}\, x}{3}, i\right )\right )}{3 \sqrt {\sqrt {3}\, \sqrt {b}}\, \sqrt {-b \,x^{4}+3}\, \sqrt {b}} \]
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 \[ \int \frac {x^{2}}{\sqrt {-b x^{4} + 3}}\,{d x} \]
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 \[ \int \frac {x^2}{\sqrt {3-b\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.39, size = 39, normalized size = 0.72 \[ \frac {\sqrt {3} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{3}} \right )}}{12 \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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