Optimal. Leaf size=23 \[ -\frac{\sqrt{b x^2+c x^4}}{b x^2} \]
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Rubi [A] time = 0.0400943, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3, 2014} \[ -\frac{\sqrt{b x^2+c x^4}}{b x^2} \]
Antiderivative was successfully verified.
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Rule 3
Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx &=\int \frac{1}{x \sqrt{b x^2+c x^4}} \, dx\\ &=-\frac{\sqrt{b x^2+c x^4}}{b x^2}\\ \end{align*}
Mathematica [A] time = 0.0070438, size = 23, normalized size = 1. \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )}}{b x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 26, normalized size = 1.1 \begin{align*} -{\frac{c{x}^{2}+b}{b}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45414, size = 41, normalized size = 1.78 \begin{align*} -\frac{\sqrt{c x^{4} + b x^{2}}}{b x^{2}} \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}{x \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13326, size = 19, normalized size = 0.83 \begin{align*} -\frac{\sqrt{c + \frac{b}{x^{2}}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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