Optimal. Leaf size=19 \[ \frac{g x}{\sqrt{a+b x^2+c x^4}} \]
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Rubi [A] time = 0.0181643, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1588} \[ \frac{g x}{\sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin{align*} \int \frac{a g-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac{g x}{\sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [F] time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [A] time = 0.006, size = 18, normalized size = 1. \begin{align*}{gx{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12618, size = 23, normalized size = 1.21 \begin{align*} \frac{g x}{\sqrt{c x^{4} + b x^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21719, size = 39, normalized size = 2.05 \begin{align*} \frac{g x}{\sqrt{c x^{4} + b x^{2} + a}} \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 [B] time = 1.21693, size = 95, normalized size = 5. \begin{align*} \frac{{\left (b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g\right )} x}{32 \,{\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} \sqrt{c x^{4} + b x^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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