3.35 \(\int \frac{x \left (2+3 x^2\right )}{\sqrt{5+x^4}} \, dx\)

Optimal. Leaf size=24 \[ \frac{3 \sqrt{x^4+5}}{2}+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

[Out]

(3*Sqrt[5 + x^4])/2 + ArcSinh[x^2/Sqrt[5]]

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Rubi [A]  time = 0.052905, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3 \sqrt{x^4+5}}{2}+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x*(2 + 3*x^2))/Sqrt[5 + x^4],x]

[Out]

(3*Sqrt[5 + x^4])/2 + ArcSinh[x^2/Sqrt[5]]

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Rubi in Sympy [A]  time = 6.57961, size = 22, normalized size = 0.92 \[ \frac{3 \sqrt{x^{4} + 5}}{2} + \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(3*x**2+2)/(x**4+5)**(1/2),x)

[Out]

3*sqrt(x**4 + 5)/2 + asinh(sqrt(5)*x**2/5)

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Mathematica [A]  time = 0.0145477, size = 24, normalized size = 1. \[ \frac{3 \sqrt{x^4+5}}{2}+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(2 + 3*x^2))/Sqrt[5 + x^4],x]

[Out]

(3*Sqrt[5 + x^4])/2 + ArcSinh[x^2/Sqrt[5]]

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Maple [A]  time = 0.014, size = 20, normalized size = 0.8 \[{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) +{\frac{3}{2}\sqrt{{x}^{4}+5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(3*x^2+2)/(x^4+5)^(1/2),x)

[Out]

arcsinh(1/5*5^(1/2)*x^2)+3/2*(x^4+5)^(1/2)

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Maxima [A]  time = 0.779643, size = 57, normalized size = 2.38 \[ \frac{3}{2} \, \sqrt{x^{4} + 5} + \frac{1}{2} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{1}{2} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x/sqrt(x^4 + 5),x, algorithm="maxima")

[Out]

3/2*sqrt(x^4 + 5) + 1/2*log(sqrt(x^4 + 5)/x^2 + 1) - 1/2*log(sqrt(x^4 + 5)/x^2 -
1)

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Fricas [A]  time = 0.280068, size = 88, normalized size = 3.67 \[ -\frac{3 \, x^{4} - 3 \, \sqrt{x^{4} + 5} x^{2} + 2 \,{\left (x^{2} - \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) + 15}{2 \,{\left (x^{2} - \sqrt{x^{4} + 5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x/sqrt(x^4 + 5),x, algorithm="fricas")

[Out]

-1/2*(3*x^4 - 3*sqrt(x^4 + 5)*x^2 + 2*(x^2 - sqrt(x^4 + 5))*log(-x^2 + sqrt(x^4
+ 5)) + 15)/(x^2 - sqrt(x^4 + 5))

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Sympy [A]  time = 2.08159, size = 22, normalized size = 0.92 \[ \frac{3 \sqrt{x^{4} + 5}}{2} + \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(3*x**2+2)/(x**4+5)**(1/2),x)

[Out]

3*sqrt(x**4 + 5)/2 + asinh(sqrt(5)*x**2/5)

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GIAC/XCAS [A]  time = 0.265035, size = 35, normalized size = 1.46 \[ \frac{3}{2} \, \sqrt{x^{4} + 5} -{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x/sqrt(x^4 + 5),x, algorithm="giac")

[Out]

3/2*sqrt(x^4 + 5) - ln(-x^2 + sqrt(x^4 + 5))