3.533 \(\int \frac{x^4}{\sqrt{9+4 x^2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{27}{128} \sqrt{4 x^2+9} x+\frac{1}{16} \sqrt{4 x^2+9} x^3+\frac{243}{256} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

(-27*x*Sqrt[9 + 4*x^2])/128 + (x^3*Sqrt[9 + 4*x^2])/16 + (243*ArcSinh[(2*x)/3])/
256

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Rubi [A]  time = 0.0391307, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{27}{128} \sqrt{4 x^2+9} x+\frac{1}{16} \sqrt{4 x^2+9} x^3+\frac{243}{256} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[9 + 4*x^2],x]

[Out]

(-27*x*Sqrt[9 + 4*x^2])/128 + (x^3*Sqrt[9 + 4*x^2])/16 + (243*ArcSinh[(2*x)/3])/
256

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Rubi in Sympy [A]  time = 5.33303, size = 39, normalized size = 0.87 \[ \frac{x^{3} \sqrt{4 x^{2} + 9}}{16} - \frac{27 x \sqrt{4 x^{2} + 9}}{128} + \frac{243 \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(4*x**2+9)**(1/2),x)

[Out]

x**3*sqrt(4*x**2 + 9)/16 - 27*x*sqrt(4*x**2 + 9)/128 + 243*asinh(2*x/3)/256

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Mathematica [A]  time = 0.0295799, size = 36, normalized size = 0.8 \[ \sqrt{4 x^2+9} \left (\frac{x^3}{16}-\frac{27 x}{128}\right )+\frac{243}{256} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[9 + 4*x^2],x]

[Out]

Sqrt[9 + 4*x^2]*((-27*x)/128 + x^3/16) + (243*ArcSinh[(2*x)/3])/256

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Maple [A]  time = 0.007, size = 34, normalized size = 0.8 \[{\frac{243}{256}{\it Arcsinh} \left ({\frac{2\,x}{3}} \right ) }-{\frac{27\,x}{128}\sqrt{4\,{x}^{2}+9}}+{\frac{{x}^{3}}{16}\sqrt{4\,{x}^{2}+9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(4*x^2+9)^(1/2),x)

[Out]

243/256*arcsinh(2/3*x)-27/128*x*(4*x^2+9)^(1/2)+1/16*x^3*(4*x^2+9)^(1/2)

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Maxima [A]  time = 1.482, size = 45, normalized size = 1. \[ \frac{1}{16} \, \sqrt{4 \, x^{2} + 9} x^{3} - \frac{27}{128} \, \sqrt{4 \, x^{2} + 9} x + \frac{243}{256} \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(4*x^2 + 9),x, algorithm="maxima")

[Out]

1/16*sqrt(4*x^2 + 9)*x^3 - 27/128*sqrt(4*x^2 + 9)*x + 243/256*arcsinh(2/3*x)

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Fricas [A]  time = 0.2331, size = 176, normalized size = 3.91 \[ -\frac{4096 \, x^{8} - 36288 \, x^{4} - 34992 \, x^{2} + 243 \,{\left (128 \, x^{4} + 288 \, x^{2} - 8 \,{\left (8 \, x^{3} + 9 \, x\right )} \sqrt{4 \, x^{2} + 9} + 81\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) - 2 \,{\left (1024 \, x^{7} - 1152 \, x^{5} - 7128 \, x^{3} - 2187 \, x\right )} \sqrt{4 \, x^{2} + 9}}{256 \,{\left (128 \, x^{4} + 288 \, x^{2} - 8 \,{\left (8 \, x^{3} + 9 \, x\right )} \sqrt{4 \, x^{2} + 9} + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(4*x^2 + 9),x, algorithm="fricas")

[Out]

-1/256*(4096*x^8 - 36288*x^4 - 34992*x^2 + 243*(128*x^4 + 288*x^2 - 8*(8*x^3 + 9
*x)*sqrt(4*x^2 + 9) + 81)*log(-2*x + sqrt(4*x^2 + 9)) - 2*(1024*x^7 - 1152*x^5 -
7128*x^3 - 2187*x)*sqrt(4*x^2 + 9))/(128*x^4 + 288*x^2 - 8*(8*x^3 + 9*x)*sqrt(4
*x^2 + 9) + 81)

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Sympy [A]  time = 1.97666, size = 39, normalized size = 0.87 \[ \frac{x^{3} \sqrt{4 x^{2} + 9}}{16} - \frac{27 x \sqrt{4 x^{2} + 9}}{128} + \frac{243 \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(4*x**2+9)**(1/2),x)

[Out]

x**3*sqrt(4*x**2 + 9)/16 - 27*x*sqrt(4*x**2 + 9)/128 + 243*asinh(2*x/3)/256

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GIAC/XCAS [A]  time = 0.212123, size = 49, normalized size = 1.09 \[ \frac{1}{128} \,{\left (8 \, x^{2} - 27\right )} \sqrt{4 \, x^{2} + 9} x - \frac{243}{256} \,{\rm ln}\left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(4*x^2 + 9),x, algorithm="giac")

[Out]

1/128*(8*x^2 - 27)*sqrt(4*x^2 + 9)*x - 243/256*ln(-2*x + sqrt(4*x^2 + 9))