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

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

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Rubi [A]  time = 0.0427033, 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

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Rubi in Sympy [A]  time = 5.538, 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

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Mathematica [A]  time = 0.00925391, 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

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Maple [A]  time = 0.005, size = 19, normalized size = 0.6 \[{\frac{2\,{x}^{2}-3}{40} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(4*x^2+9)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.49697, 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)

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Fricas [A]  time = 0.226146, 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)

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Sympy [A]  time = 1.76944, 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

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GIAC/XCAS [A]  time = 0.202521, 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)