Optimal. Leaf size=103 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}} \]
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Rubi [A] time = 0.0212074, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {398} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 398
Rubi steps
\begin{align*} \int \frac{1}{\left (-2 a-b x^2\right ) \sqrt [4]{-a-b x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}\\ \end{align*}
Mathematica [C] time = 0.155954, size = 168, normalized size = 1.63 \[ -\frac{6 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )}{\sqrt [4]{-a-b x^2} \left (2 a+b x^2\right ) \left (6 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )-b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-b{x}^{2}-2\,a}{\frac{1}{\sqrt [4]{-b{x}^{2}-a}}}}\, 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{1}{{\left (b x^{2} + 2 \, a\right )}{\left (-b x^{2} - a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{2 a \sqrt [4]{- a - b x^{2}} + b x^{2} \sqrt [4]{- a - b x^{2}}}\, 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{1}{{\left (b x^{2} + 2 \, a\right )}{\left (-b x^{2} - a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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