Optimal. Leaf size=215 \[ \frac{\sqrt{2} b \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ),-\frac{b-\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}+b}\right )}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\sqrt{b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|-\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}} \]
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Rubi [A] time = 0.207668, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 59, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {423, 424, 419} \[ \frac{\sqrt{2} b F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|-\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\sqrt{b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )|-\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}} \]
Antiderivative was successfully verified.
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Rule 423
Rule 424
Rule 419
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 b) \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}{b-\sqrt{b^2-4 a c}}-\frac{\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}{b-\sqrt{b^2-4 a c}}\\ &=-\frac{\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} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} b F\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{c} \sqrt{b-\sqrt{b^2-4 a c}}}\\ \end{align*}
Mathematica [A] time = 0.0897935, size = 102, normalized size = 0.47 \[ \frac{\sqrt{-\sqrt{b^2-4 a c}-b} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-b-\sqrt{b^2-4 a c}}}\right )|\frac{b+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}-b}\right )}{\sqrt{2} \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.115, 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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