3.805 \(\int x \sqrt{3+2 x^4} \, dx\)

Optimal. Leaf size=40 \[ \frac{3 \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x^2\right )}{4 \sqrt{2}}+\frac{1}{4} \sqrt{2 x^4+3} x^2 \]

[Out]

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

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Rubi [A]  time = 0.0399934, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x^2\right )}{4 \sqrt{2}}+\frac{1}{4} \sqrt{2 x^4+3} x^2 \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[3 + 2*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.87534, size = 34, normalized size = 0.85 \[ \frac{x^{2} \sqrt{2 x^{4} + 3}}{4} + \frac{3 \sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{6} x^{2}}{3} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(2*x**4 + 3)/4 + 3*sqrt(2)*asinh(sqrt(6)*x**2/3)/8

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Mathematica [A]  time = 0.0227729, size = 40, normalized size = 1. \[ \frac{3 \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x^2\right )}{4 \sqrt{2}}+\frac{1}{4} \sqrt{2 x^4+3} x^2 \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[3 + 2*x^4],x]

[Out]

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

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Maple [A]  time = 0.011, size = 30, normalized size = 0.8 \[{\frac{3\,\sqrt{2}}{8}{\it Arcsinh} \left ({\frac{{x}^{2}\sqrt{6}}{3}} \right ) }+{\frac{{x}^{2}}{4}\sqrt{2\,{x}^{4}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(2*x^4+3)^(1/2),x)

[Out]

3/8*arcsinh(1/3*x^2*6^(1/2))*2^(1/2)+1/4*x^2*(2*x^4+3)^(1/2)

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Maxima [A]  time = 1.58951, size = 103, normalized size = 2.58 \[ -\frac{3}{16} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \frac{\sqrt{2 \, x^{4} + 3}}{x^{2}}\right )}}{2 \, \sqrt{2} + \frac{2 \, \sqrt{2 \, x^{4} + 3}}{x^{2}}}\right ) + \frac{3 \, \sqrt{2 \, x^{4} + 3}}{4 \, x^{2}{\left (\frac{2 \, x^{4} + 3}{x^{4}} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/16*sqrt(2)*log(-2*(sqrt(2) - sqrt(2*x^4 + 3)/x^2)/((2*sqrt(2)) + 2*sqrt(2*x^4
+ 3)/x^2)) + 3/4*sqrt(2*x^4 + 3)/(x^2*((2*x^4 + 3)/x^4 - 2))

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Fricas [A]  time = 0.250966, size = 72, normalized size = 1.8 \[ \frac{1}{16} \, \sqrt{2}{\left (2 \, \sqrt{2} \sqrt{2 \, x^{4} + 3} x^{2} + 3 \, \log \left (-4 \, \sqrt{2 \, x^{4} + 3} x^{2} - \sqrt{2}{\left (4 \, x^{4} + 3\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(2)*(2*sqrt(2)*sqrt(2*x^4 + 3)*x^2 + 3*log(-4*sqrt(2*x^4 + 3)*x^2 - sqr
t(2)*(4*x^4 + 3)))

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Sympy [A]  time = 4.95882, size = 51, normalized size = 1.27 \[ \frac{x^{6}}{2 \sqrt{2 x^{4} + 3}} + \frac{3 x^{2}}{4 \sqrt{2 x^{4} + 3}} + \frac{3 \sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{6} x^{2}}{3} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**6/(2*sqrt(2*x**4 + 3)) + 3*x**2/(4*sqrt(2*x**4 + 3)) + 3*sqrt(2)*asinh(sqrt(6
)*x**2/3)/8

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GIAC/XCAS [A]  time = 0.215324, size = 53, normalized size = 1.32 \[ \frac{1}{4} \, \sqrt{2 \, x^{4} + 3} x^{2} - \frac{3}{8} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} x^{2} + \sqrt{2 \, x^{4} + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2*x^4 + 3)*x^2 - 3/8*sqrt(2)*ln(-sqrt(2)*x^2 + sqrt(2*x^4 + 3))