Optimal. Leaf size=74 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt{2} b^{3/2}} \]
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Rubi [A] time = 0.025874, antiderivative size = 74, normalized size of antiderivative = 1., 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 = {442} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt{2} b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 442
Rubi steps
\begin{align*} \int \frac{x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt{2} b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0517411, size = 55, normalized size = 0.74 \[ -\frac{x^3 \left (b x^2+1\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-b x^2,-\frac{b x^2}{2}\right )}{6 \left (-b x^2-1\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{-b{x}^{2}-2} \left ( -b{x}^{2}-1 \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} + 2\right )}{\left (-b x^{2} - 1\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 = 1.70306, size = 682, normalized size = 9.22 \begin{align*} \left [-\frac{2 \, \sqrt{2} \sqrt{b} \arctan \left (\frac{\sqrt{2}{\left (-b x^{2} - 1\right )}^{\frac{1}{4}}}{\sqrt{b} x}\right ) - \sqrt{2} \sqrt{b} \log \left (-\frac{b^{2} x^{4} + 4 \, \sqrt{-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt{2}{\left ({\left (-b x^{2} - 1\right )}^{\frac{1}{4}} b x^{3} + 2 \,{\left (-b x^{2} - 1\right )}^{\frac{3}{4}} x\right )} \sqrt{b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}, \frac{2 \, \sqrt{2} \sqrt{-b} \arctan \left (\frac{\sqrt{2}{\left (-b x^{2} - 1\right )}^{\frac{1}{4}} \sqrt{-b}}{b x}\right ) - \sqrt{2} \sqrt{-b} \log \left (-\frac{b^{2} x^{4} - 4 \, \sqrt{-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt{2}{\left ({\left (-b x^{2} - 1\right )}^{\frac{1}{4}} b x^{3} - 2 \,{\left (-b x^{2} - 1\right )}^{\frac{3}{4}} x\right )} \sqrt{-b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{4 \, b^{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}}{b x^{2} \left (- b x^{2} - 1\right )^{\frac{3}{4}} + 2 \left (- b x^{2} - 1\right )^{\frac{3}{4}}}\, 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} + 2\right )}{\left (-b x^{2} - 1\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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