Optimal. Leaf size=32 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^2}+b x^2}}\right )}{2 \sqrt{b}} \]
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Rubi [A] time = 0.0159634, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1979, 2008, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^2}+b x^2}}\right )}{2 \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 1979
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\frac{a+b x^4}{x^2}}} \, dx &=\int \frac{1}{\sqrt{\frac{a}{x^2}+b x^2}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{\frac{a}{x^2}+b x^2}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^2}+b x^2}}\right )}{2 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0207978, size = 59, normalized size = 1.84 \[ \frac{\sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 \sqrt{b} x \sqrt{\frac{a+b x^4}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 49, normalized size = 1.5 \begin{align*}{\frac{1}{2\,x}\sqrt{b{x}^{4}+a}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){\frac{1}{\sqrt{{\frac{b{x}^{4}+a}{{x}^{2}}}}}}{\frac{1}{\sqrt{b}}}} \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*} b \int \frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} + \frac{x^{2}}{2 \, \sqrt{b x^{4} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.835654, size = 196, normalized size = 6.12 \begin{align*} \left [\frac{\log \left (-2 \, b x^{4} - 2 \, \sqrt{b} x^{3} \sqrt{\frac{b x^{4} + a}{x^{2}}} - a\right )}{4 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{\sqrt{-b} x^{3} \sqrt{\frac{b x^{4} + a}{x^{2}}}}{b x^{4} + a}\right )}{2 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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