Optimal. Leaf size=101 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}} \]
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Rubi [A] time = 0.0193968, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {398} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}} \]
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 = 0.156617, size = 163, normalized size = 1.61 \[ -\frac{6 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )}{\left (2 a-b x^2\right ) \sqrt [4]{b x^2-a} \left (b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )\right )+6 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )\right )} \]
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} - a\right )}^{\frac{1}{4}}{\left (b x^{2} - 2 \, a\right )}}\,{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}{\left (- 2 a + b x^{2}\right ) \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} - a\right )}^{\frac{1}{4}}{\left (b x^{2} - 2 \, a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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