3.465 \(\int x^3 \sqrt{-9+4 x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{80} \left (4 x^2-9\right )^{5/2}+\frac{3}{16} \left (4 x^2-9\right )^{3/2} \]

[Out]

(3*(-9 + 4*x^2)^(3/2))/16 + (-9 + 4*x^2)^(5/2)/80

_______________________________________________________________________________________

Rubi [A]  time = 0.0424921, antiderivative size = 31, 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 \[ \frac{1}{80} \left (4 x^2-9\right )^{5/2}+\frac{3}{16} \left (4 x^2-9\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[-9 + 4*x^2],x]

[Out]

(3*(-9 + 4*x^2)^(3/2))/16 + (-9 + 4*x^2)^(5/2)/80

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.50386, size = 24, normalized size = 0.77 \[ \frac{\left (4 x^{2} - 9\right )^{\frac{5}{2}}}{80} + \frac{3 \left (4 x^{2} - 9\right )^{\frac{3}{2}}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(4*x**2-9)**(1/2),x)

[Out]

(4*x**2 - 9)**(5/2)/80 + 3*(4*x**2 - 9)**(3/2)/16

_______________________________________________________________________________________

Mathematica [A]  time = 0.00876401, size = 22, normalized size = 0.71 \[ \frac{1}{40} \left (2 x^2+3\right ) \left (4 x^2-9\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[-9 + 4*x^2],x]

[Out]

((3 + 2*x^2)*(-9 + 4*x^2)^(3/2))/40

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 29, normalized size = 0.9 \[{\frac{ \left ( 2\,x-3 \right ) \left ( 2\,x+3 \right ) \left ( 2\,{x}^{2}+3 \right ) }{40}\sqrt{4\,{x}^{2}-9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(4*x^2-9)^(1/2),x)

[Out]

1/40*(2*x-3)*(2*x+3)*(2*x^2+3)*(4*x^2-9)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.48749, size = 35, normalized size = 1.13 \[ \frac{1}{20} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} x^{2} + \frac{3}{40} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 - 9)*x^3,x, algorithm="maxima")

[Out]

1/20*(4*x^2 - 9)^(3/2)*x^2 + 3/40*(4*x^2 - 9)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.22268, size = 139, normalized size = 4.48 \[ -\frac{8192 \, x^{10} - 38400 \, x^{8} + 30240 \, x^{6} + 77760 \, x^{4} - 109350 \, x^{2} - 2 \,{\left (2048 \, x^{9} - 7296 \, x^{7} + 648 \, x^{5} + 17010 \, x^{3} - 10935 \, x\right )} \sqrt{4 \, x^{2} - 9} + 19683}{40 \,{\left (512 \, x^{5} - 1440 \, x^{3} -{\left (256 \, x^{4} - 432 \, x^{2} + 81\right )} \sqrt{4 \, x^{2} - 9} + 810 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 - 9)*x^3,x, algorithm="fricas")

[Out]

-1/40*(8192*x^10 - 38400*x^8 + 30240*x^6 + 77760*x^4 - 109350*x^2 - 2*(2048*x^9
- 7296*x^7 + 648*x^5 + 17010*x^3 - 10935*x)*sqrt(4*x^2 - 9) + 19683)/(512*x^5 -
1440*x^3 - (256*x^4 - 432*x^2 + 81)*sqrt(4*x^2 - 9) + 810*x)

_______________________________________________________________________________________

Sympy [A]  time = 1.77854, size = 44, normalized size = 1.42 \[ \frac{x^{4} \sqrt{4 x^{2} - 9}}{5} - \frac{3 x^{2} \sqrt{4 x^{2} - 9}}{20} - \frac{27 \sqrt{4 x^{2} - 9}}{40} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(4*x**2-9)**(1/2),x)

[Out]

x**4*sqrt(4*x**2 - 9)/5 - 3*x**2*sqrt(4*x**2 - 9)/20 - 27*sqrt(4*x**2 - 9)/40

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.2021, size = 31, normalized size = 1. \[ \frac{1}{80} \,{\left (4 \, x^{2} - 9\right )}^{\frac{5}{2}} + \frac{3}{16} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 - 9)*x^3,x, algorithm="giac")

[Out]

1/80*(4*x^2 - 9)^(5/2) + 3/16*(4*x^2 - 9)^(3/2)