3.466 \(\int x^2 \sqrt{-9+4 x^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac{9}{32} \sqrt{4 x^2-9} x-\frac{81}{64} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right )+\frac{1}{4} \sqrt{4 x^2-9} x^3 \]

[Out]

(-9*x*Sqrt[-9 + 4*x^2])/32 + (x^3*Sqrt[-9 + 4*x^2])/4 - (81*ArcTanh[(2*x)/Sqrt[-
9 + 4*x^2]])/64

_______________________________________________________________________________________

Rubi [A]  time = 0.0467754, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{9}{32} \sqrt{4 x^2-9} x-\frac{81}{64} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right )+\frac{1}{4} \sqrt{4 x^2-9} x^3 \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[-9 + 4*x^2],x]

[Out]

(-9*x*Sqrt[-9 + 4*x^2])/32 + (x^3*Sqrt[-9 + 4*x^2])/4 - (81*ArcTanh[(2*x)/Sqrt[-
9 + 4*x^2]])/64

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.14923, size = 48, normalized size = 0.89 \[ \frac{x^{3} \sqrt{4 x^{2} - 9}}{4} - \frac{9 x \sqrt{4 x^{2} - 9}}{32} - \frac{81 \operatorname{atanh}{\left (\frac{2 x}{\sqrt{4 x^{2} - 9}} \right )}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*sqrt(4*x**2 - 9)/4 - 9*x*sqrt(4*x**2 - 9)/32 - 81*atanh(2*x/sqrt(4*x**2 - 9
))/64

_______________________________________________________________________________________

Mathematica [A]  time = 0.0230525, size = 46, normalized size = 0.85 \[ \sqrt{4 x^2-9} \left (\frac{x^3}{4}-\frac{9 x}{32}\right )-\frac{81}{64} \log \left (\sqrt{4 x^2-9}+2 x\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-9 + 4*x^2]*((-9*x)/32 + x^3/4) - (81*Log[2*x + Sqrt[-9 + 4*x^2]])/64

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 47, normalized size = 0.9 \[{\frac{x}{16} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,x}{32}\sqrt{4\,{x}^{2}-9}}-{\frac{81\,\sqrt{4}}{128}\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-9} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(4*x^2-9)^(1/2),x)

[Out]

1/16*x*(4*x^2-9)^(3/2)+9/32*x*(4*x^2-9)^(1/2)-81/128*ln(x*4^(1/2)+(4*x^2-9)^(1/2
))*4^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51012, size = 58, normalized size = 1.07 \[ \frac{1}{16} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} x + \frac{9}{32} \, \sqrt{4 \, x^{2} - 9} x - \frac{81}{64} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(4*x^2 - 9)^(3/2)*x + 9/32*sqrt(4*x^2 - 9)*x - 81/64*log(8*x + 4*sqrt(4*x^2
- 9))

_______________________________________________________________________________________

Fricas [A]  time = 0.22284, size = 182, normalized size = 3.37 \[ -\frac{4096 \, x^{8} - 18432 \, x^{6} + 25920 \, x^{4} - 11664 \, x^{2} - 81 \,{\left (128 \, x^{4} - 288 \, x^{2} - 8 \,{\left (8 \, x^{3} - 9 \, x\right )} \sqrt{4 \, x^{2} - 9} + 81\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} - 9}\right ) - 2 \,{\left (1024 \, x^{7} - 3456 \, x^{5} + 3240 \, x^{3} - 729 \, x\right )} \sqrt{4 \, x^{2} - 9}}{64 \,{\left (128 \, x^{4} - 288 \, x^{2} - 8 \,{\left (8 \, x^{3} - 9 \, x\right )} \sqrt{4 \, x^{2} - 9} + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(4096*x^8 - 18432*x^6 + 25920*x^4 - 11664*x^2 - 81*(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 -
3456*x^5 + 3240*x^3 - 729*x)*sqrt(4*x^2 - 9))/(128*x^4 - 288*x^2 - 8*(8*x^3 - 9*
x)*sqrt(4*x^2 - 9) + 81)

_______________________________________________________________________________________

Sympy [A]  time = 9.21466, size = 124, normalized size = 2.3 \[ \begin{cases} \frac{x^{5}}{\sqrt{4 x^{2} - 9}} - \frac{27 x^{3}}{8 \sqrt{4 x^{2} - 9}} + \frac{81 x}{32 \sqrt{4 x^{2} - 9}} - \frac{81 \operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{64} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\- \frac{i x^{5}}{\sqrt{- 4 x^{2} + 9}} + \frac{27 i x^{3}}{8 \sqrt{- 4 x^{2} + 9}} - \frac{81 i x}{32 \sqrt{- 4 x^{2} + 9}} + \frac{81 i \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{64} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((x**5/sqrt(4*x**2 - 9) - 27*x**3/(8*sqrt(4*x**2 - 9)) + 81*x/(32*sqrt(
4*x**2 - 9)) - 81*acosh(2*x/3)/64, 4*Abs(x**2)/9 > 1), (-I*x**5/sqrt(-4*x**2 + 9
) + 27*I*x**3/(8*sqrt(-4*x**2 + 9)) - 81*I*x/(32*sqrt(-4*x**2 + 9)) + 81*I*asin(
2*x/3)/64, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205321, size = 50, normalized size = 0.93 \[ \frac{1}{32} \,{\left (8 \, x^{2} - 9\right )} \sqrt{4 \, x^{2} - 9} x + \frac{81}{64} \,{\rm ln}\left ({\left | -2 \, x + \sqrt{4 \, x^{2} - 9} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(8*x^2 - 9)*sqrt(4*x^2 - 9)*x + 81/64*ln(abs(-2*x + sqrt(4*x^2 - 9)))