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 = 0.030975, antiderivative size = 124, normalized size of antiderivative = 1., 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*}
Mathematica [C] time = 0.045506, 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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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{b{x}^{2}+4} \left ( b{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (b x^{2} + 4\right )}{\left (b x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.93609, size = 1249, normalized size = 10.07 \begin{align*} \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} x \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}} - 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (b x^{2} + 2\right )^{\frac{3}{4}} \left (b x^{2} + 4\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (b x^{2} + 4\right )}{\left (b x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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