Optimal. Leaf size=124 \[ \frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {b x^2+2}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {b x^2+2}+2\ 2^{3/4}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{\sqrt [4]{2} b^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {441} \[ \frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {b x^2+2}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {b x^2+2}+2\ 2^{3/4}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{\sqrt [4]{2} b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 441
Rubi steps
\begin {align*} \int \frac {x^2}{\left (2+b x^2\right )^{3/4} \left (4+b x^2\right )} \, dx &=-\frac {\tan ^{-1}\left (\frac {2\ 2^{3/4}+2 \sqrt [4]{2} \sqrt {2+b x^2}}{2 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}+\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {2+b x^2}}{2 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 39, normalized size = 0.31 \[ \frac {x^3 F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};-\frac {b x^2}{2},-\frac {b x^2}{4}\right )}{12\ 2^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.04, size = 393, normalized size = 3.17 \[ \sqrt {2} \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{b^{6}}^{\frac {1}{4}} \arctan \left (\frac {8 \, \sqrt {2} \sqrt {\frac {1}{2}} \left (\frac {1}{8}\right )^{\frac {3}{4}} b^{4} \sqrt {\frac {\sqrt {\frac {1}{2}} b^{4} \sqrt {\frac {1}{b^{6}}} x^{2} + 2 \, \sqrt {2} \left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{6}}^{\frac {1}{4}} x + 2 \, \sqrt {b x^{2} + 2}}{x^{2}}} \frac {1}{b^{6}}^{\frac {3}{4}} x - 8 \, \sqrt {2} \left (\frac {1}{8}\right )^{\frac {3}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{4} \frac {1}{b^{6}}^{\frac {3}{4}} - x}{x}\right ) + \sqrt {2} \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{b^{6}}^{\frac {1}{4}} \arctan \left (\frac {8 \, \sqrt {2} \sqrt {\frac {1}{2}} \left (\frac {1}{8}\right )^{\frac {3}{4}} b^{4} \sqrt {\frac {\sqrt {\frac {1}{2}} b^{4} \sqrt {\frac {1}{b^{6}}} x^{2} - 2 \, \sqrt {2} \left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{6}}^{\frac {1}{4}} x + 2 \, \sqrt {b x^{2} + 2}}{x^{2}}} \frac {1}{b^{6}}^{\frac {3}{4}} x - 8 \, \sqrt {2} \left (\frac {1}{8}\right )^{\frac {3}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{4} \frac {1}{b^{6}}^{\frac {3}{4}} + x}{x}\right ) - \frac {1}{4} \, \sqrt {2} \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{b^{6}}^{\frac {1}{4}} \log \left (\frac {\sqrt {\frac {1}{2}} b^{4} \sqrt {\frac {1}{b^{6}}} x^{2} + 2 \, \sqrt {2} \left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{6}}^{\frac {1}{4}} x + 2 \, \sqrt {b x^{2} + 2}}{2 \, x^{2}}\right ) + \frac {1}{4} \, \sqrt {2} \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{b^{6}}^{\frac {1}{4}} \log \left (\frac {\sqrt {\frac {1}{2}} b^{4} \sqrt {\frac {1}{b^{6}}} x^{2} - 2 \, \sqrt {2} \left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{6}}^{\frac {1}{4}} x + 2 \, \sqrt {b x^{2} + 2}}{2 \, x^{2}}\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 \[ \int \frac {x^{2}}{{\left (b x^{2} + 4\right )} {\left (b x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (b \,x^{2}+2\right )^{\frac {3}{4}} \left (b \,x^{2}+4\right )}\, dx \]
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 \[ \int \frac {x^{2}}{{\left (b x^{2} + 4\right )} {\left (b x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\left (b\,x^2+2\right )}^{3/4}\,\left (b\,x^2+4\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (b x^{2} + 2\right )^{\frac {3}{4}} \left (b x^{2} + 4\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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