Optimal. Leaf size=47 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}+b x^2}}\right )}{\sqrt{2} \sqrt{b}} \]
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Rubi [A] time = 0.10908, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {2132, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}+b x^2}}\right )}{\sqrt{2} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 2132
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{b x^2+\sqrt{a+b^2 x^4}}}{\sqrt{a+b^2 x^4}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-2 b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+\sqrt{a+b^2 x^4}}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{b x^2+\sqrt{a+b^2 x^4}}}\right )}{\sqrt{2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.022504, size = 47, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}+b x^2}}\right )}{\sqrt{2} \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{b{x}^{2}+\sqrt{{b}^{2}{x}^{4}+a}}{\frac{1}{\sqrt{{b}^{2}{x}^{4}+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{\sqrt{b x^{2} + \sqrt{b^{2} x^{4} + a}}}{\sqrt{b^{2} x^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 12.0447, size = 348, normalized size = 7.4 \begin{align*} \left [\frac{\sqrt{2} \log \left (4 \, b^{2} x^{4} + 4 \, \sqrt{b^{2} x^{4} + a} b x^{2} + 2 \,{\left (\sqrt{2} b^{\frac{3}{2}} x^{3} + \sqrt{2} \sqrt{b^{2} x^{4} + a} \sqrt{b} x\right )} \sqrt{b x^{2} + \sqrt{b^{2} x^{4} + a}} + a\right )}{4 \, \sqrt{b}}, -\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{b}} \arctan \left (\frac{\sqrt{2} \sqrt{b x^{2} + \sqrt{b^{2} x^{4} + a}} \sqrt{-\frac{1}{b}}}{2 \, x}\right )\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{\sqrt{b x^{2} + \sqrt{a + b^{2} x^{4}}}}{\sqrt{a + b^{2} x^{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{\sqrt{b x^{2} + \sqrt{b^{2} x^{4} + a}}}{\sqrt{b^{2} x^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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