Optimal. Leaf size=49 \[ -\frac{\tanh ^{-1}\left (\frac{x \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x^2+b x^4+c x^6}}\right )}{2 \sqrt{a}} \]
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Rubi [A] time = 0.0151489, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1996, 1904, 206} \[ -\frac{\tanh ^{-1}\left (\frac{x \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x^2+b x^4+c x^6}}\right )}{2 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 1996
Rule 1904
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x^2 \left (a+b x^2+c x^4\right )}} \, dx &=\int \frac{1}{\sqrt{a x^2+b x^4+c x^6}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{x \left (2 a+b x^2\right )}{\sqrt{a x^2+b x^4+c x^6}}\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{x \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x^2+b x^4+c x^6}}\right )}{2 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0163743, size = 81, normalized size = 1.65 \[ -\frac{x \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a} \sqrt{x^2 \left (a+b x^2+c x^4\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 72, normalized size = 1.5 \begin{align*} -{\frac{x}{2}\sqrt{c{x}^{4}+b{x}^{2}+a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{{x}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}}{\frac{1}{\sqrt{a}}}} \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}{\sqrt{{\left (c x^{4} + b x^{2} + a\right )} x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32455, size = 311, normalized size = 6.35 \begin{align*} \left [\frac{\log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{6} + b x^{4} + a x^{2}}{\left (b x^{2} + 2 \, a\right )} \sqrt{a}}{x^{5}}\right )}{4 \, \sqrt{a}}, \frac{\sqrt{-a} \arctan \left (\frac{\sqrt{c x^{6} + b x^{4} + a x^{2}}{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{5} + a b x^{3} + a^{2} x\right )}}\right )}{2 \, a}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (c x^{4} + b x^{2} + a\right )} x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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