Optimal. Leaf size=478 \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ),-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{x \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}}{\sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}-\frac{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} E\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.36674, antiderivative size = 478, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 59, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068, Rules used = {422, 418, 492, 411} \[ \frac{x \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}}{\sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}-\frac{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} E\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1}{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 422
Rule 418
Rule 492
Rule 411
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{(2 c) \int \frac{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}{b-\sqrt{b^2-4 a c}}+\int \frac{1}{\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{x \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}}}}+\frac{\sqrt{b+\sqrt{b^2-4 a c}} \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}{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}}}}-\int \frac{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}}{\left (1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )^{3/2}} \, dx\\ &=\frac{x \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}}}}-\frac{\sqrt{b+\sqrt{b^2-4 a c}} \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}{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}}}}+\frac{\sqrt{b+\sqrt{b^2-4 a c}} \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )|-\frac{2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}}{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}}}}\\ \end{align*}
Mathematica [A] time = 0.116894, size = 102, normalized size = 0.21 \[ \frac{\sqrt{-\sqrt{b^2-4 a c}-b} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-b-\sqrt{b^2-4 a c}}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.159, 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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