Optimal. Leaf size=278 \[ \frac{x \left (2 x^2+\sqrt{13}+5\right )}{3 \sqrt{x^4+5 x^2+3}}-\frac{2 \sqrt{x^4+5 x^2+3}}{3 x}+\frac{\sqrt{\frac{3}{2 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{x^4+5 x^2+3}}-\frac{\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{3 \sqrt{x^4+5 x^2+3}} \]
[Out]
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Rubi [A] time = 0.302223, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{x \left (2 x^2+\sqrt{13}+5\right )}{3 \sqrt{x^4+5 x^2+3}}-\frac{2 \sqrt{x^4+5 x^2+3}}{3 x}+\frac{\sqrt{\frac{3}{2 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{x^4+5 x^2+3}}-\frac{\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{3 \sqrt{x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x^2)/(x^2*Sqrt[3 + 5*x^2 + x^4]),x]
[Out]
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Rubi in Sympy [A] time = 24.1463, size = 250, normalized size = 0.9 \[ \frac{x \left (2 x^{2} + \sqrt{13} + 5\right )}{3 \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{\sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \sqrt{\sqrt{13} + 5} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{18 \sqrt{x^{4} + 5 x^{2} + 3}} + \frac{\sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{2 \sqrt{\sqrt{13} + 5} \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{2 \sqrt{x^{4} + 5 x^{2} + 3}}{3 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3*x**2+2)/x**2/(x**4+5*x**2+3)**(1/2),x)
[Out]
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Mathematica [C] time = 0.53472, size = 224, normalized size = 0.81 \[ \frac{-i \sqrt{2} \left (4+\sqrt{13}\right ) x \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+i \sqrt{2} \left (\sqrt{13}-5\right ) x \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )-4 \left (x^4+5 x^2+3\right )}{6 x \sqrt{x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(2 + 3*x^2)/(x^2*Sqrt[3 + 5*x^2 + x^4]),x]
[Out]
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Maple [A] time = 0.024, size = 211, normalized size = 0.8 \[ 18\,{\frac{\sqrt{1- \left ( -5/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -5/6-1/6\,\sqrt{13} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) }{\sqrt{-30+6\,\sqrt{13}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}}-{\frac{2}{3\,x}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-24\,{\frac{\sqrt{1- \left ( -5/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -5/6-1/6\,\sqrt{13} \right ){x}^{2}} \left ({\it EllipticF} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) -{\it EllipticE} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) \right ) }{\sqrt{-30+6\,\sqrt{13}}\sqrt{{x}^{4}+5\,{x}^{2}+3} \left ( 5+\sqrt{13} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3*x^2+2)/x^2/(x^4+5*x^2+3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{3 \, x^{2} + 2}{\sqrt{x^{4} + 5 \, x^{2} + 3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)/(sqrt(x^4 + 5*x^2 + 3)*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{3 \, x^{2} + 2}{\sqrt{x^{4} + 5 \, x^{2} + 3} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)/(sqrt(x^4 + 5*x^2 + 3)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{3 x^{2} + 2}{x^{2} \sqrt{x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x**2+2)/x**2/(x**4+5*x**2+3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{3 \, x^{2} + 2}{\sqrt{x^{4} + 5 \, x^{2} + 3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)/(sqrt(x^4 + 5*x^2 + 3)*x^2),x, algorithm="giac")
[Out]