Optimal. Leaf size=35 \[ \frac{3}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{3 x^2+2}{2 \sqrt{x^4+5}} \]
[Out]
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Rubi [A] time = 0.0831988, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{3}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{3 x^2+2}{2 \sqrt{x^4+5}} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(2 + 3*x^2))/(5 + x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.20522, size = 31, normalized size = 0.89 \[ - \frac{15 x^{2} + 10}{10 \sqrt{x^{4} + 5}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(3*x**2+2)/(x**4+5)**(3/2),x)
[Out]
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Mathematica [A] time = 0.028633, size = 35, normalized size = 1. \[ \frac{3}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{-3 x^2-2}{2 \sqrt{x^4+5}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(2 + 3*x^2))/(5 + x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 34, normalized size = 1. \[ -{\frac{1}{\sqrt{{x}^{4}+5}}}-{\frac{3\,{x}^{2}}{2}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{3}{2}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(3*x^2+2)/(x^4+5)^(3/2),x)
[Out]
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Maxima [A] time = 0.787725, size = 73, normalized size = 2.09 \[ -\frac{3 \, x^{2}}{2 \, \sqrt{x^{4} + 5}} - \frac{1}{\sqrt{x^{4} + 5}} + \frac{3}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{3}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)*x^3/(x^4 + 5)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.306065, size = 95, normalized size = 2.71 \[ \frac{2 \, x^{2} - 3 \,{\left (x^{4} - \sqrt{x^{4} + 5} x^{2} + 5\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) - 2 \, \sqrt{x^{4} + 5} - 15}{2 \,{\left (x^{4} - \sqrt{x^{4} + 5} x^{2} + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)*x^3/(x^4 + 5)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.0662, size = 39, normalized size = 1.11 \[ - \frac{3 x^{2}}{2 \sqrt{x^{4} + 5}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} - \frac{1}{\sqrt{x^{4} + 5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(3*x**2+2)/(x**4+5)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.274721, size = 45, normalized size = 1.29 \[ -\frac{3 \, x^{2} + 2}{2 \, \sqrt{x^{4} + 5}} - \frac{3}{2} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)*x^3/(x^4 + 5)^(3/2),x, algorithm="giac")
[Out]