Optimal. Leaf size=113 \[ -\frac{\sqrt{b-\sqrt{b^2-4 a c}} \left (\sqrt{b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]
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Rubi [A] time = 0.203864, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 87, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023, Rules used = {21, 424} \[ -\frac{\sqrt{b-\sqrt{b^2-4 a c}} \left (\sqrt{b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 424
Rubi steps
\begin{align*} \int \frac{-b-\sqrt{b^2-4 a c}+2 c x^2}{\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 &=\left (-b-\sqrt{b^2-4 a c}\right ) \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}} \left (b+\sqrt{b^2-4 a c}\right ) 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 [C] time = 0.383008, size = 104, normalized size = 0.92 \[ -2 i \sqrt{2} a \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.236, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 2\,c{x}^{2}-\sqrt{-4\,ac+{b}^{2}}-b \right ){\frac{1}{\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{2 \, c x^{2} - b - \sqrt{b^{2} - 4 \, a c}}{\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 (2 \, a c x^{2} - a b - \sqrt{b^{2} - 4 \, a c} 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}}}{2 \,{\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{- b + 2 c x^{2} - \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}}}} \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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