Optimal. Leaf size=119 \[ \frac {\tan ^{-1}\left (\frac {2-\sqrt {2} \sqrt {2-b x^2}}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {2-b x^2}+2}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {441} \begin {gather*} \frac {\tan ^{-1}\left (\frac {2-\sqrt {2} \sqrt {2-b x^2}}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {2-b x^2}+2}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}} \end {gather*}
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-\sqrt {2} \sqrt {2-b x^2}}{\sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {2+\sqrt {2} \sqrt {2-b x^2}}{\sqrt [4]{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 = 0.33 \begin {gather*} \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}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 2.31, size = 140, normalized size = 1.18 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [4]{2} \sqrt {2-b x^2}}{\sqrt {b}}-\frac {\sqrt {b} x^2}{2 \sqrt [4]{2}}}{x \sqrt [4]{2-b x^2}}\right )}{2 \sqrt [4]{2} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {b} x \sqrt [4]{2-b x^2}}{b x^2+2 \sqrt {2} \sqrt {2-b x^2}}\right )}{2 \sqrt [4]{2} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 403, normalized size = 3.39 \begin {gather*} \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} \frac {1}{b^{6}}^{\frac {3}{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}}} - 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 {gather*}
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 \begin {gather*} \int -\frac {x^{2}}{{\left (b x^{2} - 4\right )} {\left (-b x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (-b \,x^{2}+2\right )^{\frac {3}{4}} \left (-b \,x^{2}+4\right )}\, dx \end {gather*}
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 \begin {gather*} -\int \frac {x^{2}}{{\left (b x^{2} - 4\right )} {\left (-b x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
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 \begin {gather*} -\int \frac {x^2}{{\left (2-b\,x^2\right )}^{3/4}\,\left (b\,x^2-4\right )} \,d x \end {gather*}
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 \begin {gather*} - \int \frac {x^{2}}{b x^{2} \left (- b x^{2} + 2\right )^{\frac {3}{4}} - 4 \left (- b x^{2} + 2\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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