Optimal. Leaf size=60 \[ \frac{\sqrt{a} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.0122802, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1089, 215} \[ \frac{\sqrt{a} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1089
Rule 215
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{\sqrt{1+\frac{b x^2}{a}} \int \frac{1}{\sqrt{1+\frac{b x^2}{a}}} \, dx}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\sqrt{a} \sqrt{1+\frac{b x^2}{a}} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0149033, size = 49, normalized size = 0.82 \[ \frac{\sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b} \sqrt [4]{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.2, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [4]{{b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2}}}}\, 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^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28972, size = 217, normalized size = 3.62 \begin{align*} \left [\frac{\log \left (-2 \, b x^{2} - 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} \sqrt{b} x - a\right )}{2 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} \sqrt{-b} x}{b x^{2} + a}\right )}{b}\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{1}{\sqrt [4]{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15798, size = 36, normalized size = 0.6 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{-b x^{2} - a}{\left | x \right |}}{\sqrt{b} x^{2}}\right )}{\sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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