Optimal. Leaf size=56 \[ \frac{149}{16} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{1}{8} \left (37-6 x^2\right ) \sqrt{x^4+5 x^2+3} \]
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Rubi [A] time = 0.12406, antiderivative size = 56, 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 \[ \frac{149}{16} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{1}{8} \left (37-6 x^2\right ) \sqrt{x^4+5 x^2+3} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(2 + 3*x^2))/Sqrt[3 + 5*x^2 + x^4],x]
[Out]
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Rubi in Sympy [A] time = 14.4906, size = 49, normalized size = 0.88 \[ - \frac{\left (- 3 x^{2} + \frac{37}{2}\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{4} + \frac{149 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(3*x**2+2)/(x**4+5*x**2+3)**(1/2),x)
[Out]
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Mathematica [A] time = 0.025046, size = 54, normalized size = 0.96 \[ \frac{1}{8} \sqrt{x^4+5 x^2+3} \left (6 x^2-37\right )+\frac{149}{16} \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(2 + 3*x^2))/Sqrt[3 + 5*x^2 + x^4],x]
[Out]
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Maple [A] time = 0.017, size = 53, normalized size = 1. \[ -{\frac{37}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{149}{16}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }+{\frac{3\,{x}^{2}}{4}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(3*x^2+2)/(x^4+5*x^2+3)^(1/2),x)
[Out]
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Maxima [A] time = 0.703414, size = 76, normalized size = 1.36 \[ \frac{3}{4} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} - \frac{37}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{149}{16} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)*x^3/sqrt(x^4 + 5*x^2 + 3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271298, size = 198, normalized size = 3.54 \[ -\frac{384 \, x^{8} + 512 \, x^{6} - 8248 \, x^{4} - 16024 \, x^{2} + 596 \,{\left (8 \, x^{4} + 40 \, x^{2} - 4 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \, x^{2} + 5\right )} + 37\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) - 4 \,{\left (96 \, x^{6} - 112 \, x^{4} - 1626 \, x^{2} - 513\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - 1295}{64 \,{\left (8 \, x^{4} + 40 \, x^{2} - 4 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \, x^{2} + 5\right )} + 37\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)*x^3/sqrt(x^4 + 5*x^2 + 3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (3 x^{2} + 2\right )}{\sqrt{x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(3*x**2+2)/(x**4+5*x**2+3)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.277405, size = 62, normalized size = 1.11 \[ \frac{1}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (6 \, x^{2} - 37\right )} - \frac{149}{16} \,{\rm ln}\left (2 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)*x^3/sqrt(x^4 + 5*x^2 + 3),x, algorithm="giac")
[Out]