Optimal. Leaf size=95 \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]
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Rubi [A] time = 0.132217, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 59, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.017, Rules used = {424} \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}} \, dx &=\frac{\sqrt{b+\sqrt{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.107819, size = 95, normalized size = 1. \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.23, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{1+2\,{\frac{c{x}^{2}}{b-\sqrt{-4\,ac+{b}^{2}}}}}{\frac{1}{\sqrt{1-2\,{\frac{c{x}^{2}}{b+\sqrt{-4\,ac+{b}^{2}}}}}}}}\, dx \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{\sqrt{\frac{2 \, c x^{2}}{b - \sqrt{b^{2} - 4 \, a c}} + 1}}{\sqrt{-\frac{2 \, c x^{2}}{b + \sqrt{b^{2} - 4 \, a c}} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (b x^{2} + \sqrt{b^{2} - 4 \, a c} x^{2} - 2 \, a\right )} \sqrt{\frac{b x^{2} + \sqrt{b^{2} - 4 \, a c} x^{2} + 2 \, a}{a}} \sqrt{-\frac{b x^{2} - \sqrt{b^{2} - 4 \, a c} x^{2} - 2 \, a}{a}}}{4 \,{\left (c x^{4} - b x^{2} + a\right )}}, x\right ) \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{\sqrt{\frac{b + 2 c x^{2} - \sqrt{- 4 a c + b^{2}}}{b - \sqrt{- 4 a c + b^{2}}}}}{\sqrt{- \frac{- b + 2 c x^{2} - \sqrt{- 4 a c + b^{2}}}{b + \sqrt{- 4 a c + b^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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