Optimal. Leaf size=22 \[ \frac{\sqrt{b x^2+c x^4}}{c x} \]
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Rubi [A] time = 0.0174382, antiderivative size = 22, 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, 1588} \[ \frac{\sqrt{b x^2+c x^4}}{c x} \]
Antiderivative was successfully verified.
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Rule 3
Rule 1588
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx &=\int \frac{x^2}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{\sqrt{b x^2+c x^4}}{c x}\\ \end{align*}
Mathematica [A] time = 0.0053655, size = 22, normalized size = 1. \[ \frac{\sqrt{x^2 \left (b+c x^2\right )}}{c x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 26, normalized size = 1.2 \begin{align*}{\frac{x \left ( c{x}^{2}+b \right ) }{c}{\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 [A] time = 0.99111, size = 18, normalized size = 0.82 \begin{align*} \frac{\sqrt{c x^{2} + b}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45743, size = 36, normalized size = 1.64 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}}{c x} \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{x^{2}}{\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 = 1.14299, size = 42, normalized size = 1.91 \begin{align*} -\frac{2 \, \sqrt{b}}{{\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2} - c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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