Optimal. Leaf size=240 \[ \frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]
[Out]
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Rubi [A] time = 0.925522, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x^2]/(a + b*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 103.247, size = 219, normalized size = 0.91 \[ \frac{\sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}} \operatorname{atanh}{\left (\frac{x \sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{d + e x^{2}}} \right )}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} \operatorname{atanh}{\left (\frac{x \sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{d + e x^{2}}} \right )}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d)**(1/2)/(c*x**4+b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.125985, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[d + e*x^2]/(a + b*x^2 + c*x^4),x]
[Out]
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Maple [C] time = 0.02, size = 161, normalized size = 0.7 \[ -{\frac{1}{2}{e}^{{\frac{3}{2}}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{{{\it \_R}}^{2}+2\,{\it \_R}\,d+{d}^{2}}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x^{2} + d}}{c x^{4} + b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x^2 + d)/(c*x^4 + b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.04514, size = 1330, normalized size = 5.54 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x^2 + d)/(c*x^4 + b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)**(1/2)/(c*x**4+b*x**2+a),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x^2 + d)/(c*x^4 + b*x^2 + a),x, algorithm="giac")
[Out]