Optimal. Leaf size=204 \[ \frac{\sqrt{\sqrt{4 a c+b^2}-b} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1} \Pi \left (\frac{\left (b-\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2+4 a c}-b}}\right )|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d \sqrt{-a+b x^2+c x^4}} \]
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Rubi [A] time = 0.191646, antiderivative size = 204, 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 = {1220, 537} \[ \frac{\sqrt{\sqrt{4 a c+b^2}-b} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1} \Pi \left (\frac{\left (b-\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2+4 a c}-b}}\right )|\frac{b-\sqrt{b^2+4 a c}}{b+\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d \sqrt{-a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1220
Rule 537
Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^2\right ) \sqrt{-a+b x^2+c x^4}} \, dx &=\frac{\left (\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}}}\right ) \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}}} \left (d+e x^2\right )} \, dx}{\sqrt{-a+b x^2+c x^4}}\\ &=\frac{\sqrt{-b+\sqrt{b^2+4 a c}} \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}}} \Pi \left (\frac{\left (b-\sqrt{b^2+4 a c}\right ) e}{2 c d};\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} d \sqrt{-a+b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.223136, size = 216, normalized size = 1.06 \[ -\frac{i \sqrt{\frac{\sqrt{4 a c+b^2}+b+2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \Pi \left (\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} d \sqrt{\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{-a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 198, normalized size = 1. \begin{align*}{\frac{1}{d}\sqrt{1-{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{4\,ac+{b}^{2}}}}\sqrt{1-{\frac{b{x}^{2}}{2\,a}}-{\frac{{x}^{2}}{2\,a}\sqrt{4\,ac+{b}^{2}}}}{\it EllipticPi} \left ( \sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}x,2\,{\frac{ae}{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) d}},{\frac{\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{{\frac{b}{2\,a}}-{\frac{1}{2\,a}\sqrt{4\,ac+{b}^{2}}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}-a}}}} \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{1}{\sqrt{c x^{4} + b x^{2} - a}{\left (e x^{2} + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x^{2}\right ) \sqrt{- a + b x^{2} + c x^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{4} + b x^{2} - a}{\left (e x^{2} + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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