Optimal. Leaf size=197 \[ \frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4}} \]
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Rubi [A] time = 0.523005, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x^2)*Sqrt[a + b*x^2 - c*x^4]),x]
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Rubi in Sympy [A] time = 61.4541, size = 173, normalized size = 0.88 \[ \frac{\sqrt{2} \sqrt{b + \sqrt{4 a c + b^{2}}} \sqrt{- \frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1} \sqrt{- \frac{2 c x^{2}}{b + \sqrt{4 a c + b^{2}}} + 1} \Pi \left (- \frac{e \left (b + \sqrt{4 a c + b^{2}}\right )}{2 c d}; \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{4 a c + b^{2}}}} \right )}\middle | \frac{b + \sqrt{4 a c + b^{2}}}{b - \sqrt{4 a c + b^{2}}}\right )}{2 \sqrt{c} d \sqrt{a + b x^{2} - c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x**2+d)/(-c*x**4+b*x**2+a)**(1/2),x)
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Mathematica [C] time = 0.222401, size = 205, normalized size = 1.04 \[ -\frac{i \sqrt{\frac{2 c x^2}{\sqrt{4 a c+b^2}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|-\frac{b+\sqrt{b^2+4 a c}}{\sqrt{b^2+4 a c}-b}\right )}{\sqrt{2} d \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x^2)*Sqrt[a + b*x^2 - c*x^4]),x]
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Maple [A] time = 0.038, size = 201, normalized size = 1. \[{\frac{\sqrt{2}}{d}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}-{\frac{{x}^{2}}{2\,a}\sqrt{4\,ac+{b}^{2}}}}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}},-2\,{\frac{ae}{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) d}},{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}+{\frac{1}{a}\sqrt{4\,ac+{b}^{2}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x^2+d)/(-c*x^4+b*x^2+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x^{2}\right ) \sqrt{a + b x^{2} - c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x**2+d)/(-c*x**4+b*x**2+a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="giac")
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