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

Optimal. Leaf size=72 \[ -\frac{81}{256} \sqrt{4 x^2-9} x-\frac{729}{512} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right )+\frac{1}{6} \sqrt{4 x^2-9} x^5-\frac{3}{32} \sqrt{4 x^2-9} x^3 \]

[Out]

(-81*x*Sqrt[-9 + 4*x^2])/256 - (3*x^3*Sqrt[-9 + 4*x^2])/32 + (x^5*Sqrt[-9 + 4*x^
2])/6 - (729*ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]])/512

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Rubi [A]  time = 0.0653834, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{81}{256} \sqrt{4 x^2-9} x-\frac{729}{512} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right )+\frac{1}{6} \sqrt{4 x^2-9} x^5-\frac{3}{32} \sqrt{4 x^2-9} x^3 \]

Antiderivative was successfully verified.

[In]  Int[x^4*Sqrt[-9 + 4*x^2],x]

[Out]

(-81*x*Sqrt[-9 + 4*x^2])/256 - (3*x^3*Sqrt[-9 + 4*x^2])/32 + (x^5*Sqrt[-9 + 4*x^
2])/6 - (729*ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]])/512

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Rubi in Sympy [A]  time = 8.20423, size = 65, normalized size = 0.9 \[ \frac{x^{5} \sqrt{4 x^{2} - 9}}{6} - \frac{3 x^{3} \sqrt{4 x^{2} - 9}}{32} - \frac{81 x \sqrt{4 x^{2} - 9}}{256} - \frac{729 \operatorname{atanh}{\left (\frac{2 x}{\sqrt{4 x^{2} - 9}} \right )}}{512} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*sqrt(4*x**2 - 9)/6 - 3*x**3*sqrt(4*x**2 - 9)/32 - 81*x*sqrt(4*x**2 - 9)/256
- 729*atanh(2*x/sqrt(4*x**2 - 9))/512

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Mathematica [A]  time = 0.0206437, size = 49, normalized size = 0.68 \[ \frac{1}{768} x \sqrt{4 x^2-9} \left (128 x^4-72 x^2-243\right )-\frac{729}{512} \log \left (\sqrt{4 x^2-9}+2 x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[-9 + 4*x^2]*(-243 - 72*x^2 + 128*x^4))/768 - (729*Log[2*x + Sqrt[-9 + 4*
x^2]])/512

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Maple [A]  time = 0.009, size = 61, normalized size = 0.9 \[{\frac{{x}^{3}}{24} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,x}{128} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{81\,x}{256}\sqrt{4\,{x}^{2}-9}}-{\frac{729\,\sqrt{4}}{1024}\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-9} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(4*x^2-9)^(1/2),x)

[Out]

1/24*x^3*(4*x^2-9)^(3/2)+9/128*x*(4*x^2-9)^(3/2)+81/256*x*(4*x^2-9)^(1/2)-729/10
24*ln(x*4^(1/2)+(4*x^2-9)^(1/2))*4^(1/2)

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Maxima [A]  time = 1.47962, size = 77, normalized size = 1.07 \[ \frac{1}{24} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} x^{3} + \frac{9}{128} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} x + \frac{81}{256} \, \sqrt{4 \, x^{2} - 9} x - \frac{729}{512} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(4*x^2 - 9)^(3/2)*x^3 + 9/128*(4*x^2 - 9)^(3/2)*x + 81/256*sqrt(4*x^2 - 9)*
x - 729/512*log(8*x + 4*sqrt(4*x^2 - 9))

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Fricas [A]  time = 0.228333, size = 236, normalized size = 3.28 \[ -\frac{1048576 \, x^{12} - 5308416 \, x^{10} + 6967296 \, x^{8} + 3172608 \, x^{6} - 10707552 \, x^{4} + 4251528 \, x^{2} - 2187 \,{\left (2048 \, x^{6} - 6912 \, x^{4} + 5832 \, x^{2} - 4 \,{\left (256 \, x^{5} - 576 \, x^{3} + 243 \, x\right )} \sqrt{4 \, x^{2} - 9} - 729\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} - 9}\right ) - 2 \,{\left (262144 \, x^{11} - 1032192 \, x^{9} + 746496 \, x^{7} + 1166400 \, x^{5} - 1364688 \, x^{3} + 177147 \, x\right )} \sqrt{4 \, x^{2} - 9}}{1536 \,{\left (2048 \, x^{6} - 6912 \, x^{4} + 5832 \, x^{2} - 4 \,{\left (256 \, x^{5} - 576 \, x^{3} + 243 \, x\right )} \sqrt{4 \, x^{2} - 9} - 729\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1536*(1048576*x^12 - 5308416*x^10 + 6967296*x^8 + 3172608*x^6 - 10707552*x^4
+ 4251528*x^2 - 2187*(2048*x^6 - 6912*x^4 + 5832*x^2 - 4*(256*x^5 - 576*x^3 + 24
3*x)*sqrt(4*x^2 - 9) - 729)*log(-2*x + sqrt(4*x^2 - 9)) - 2*(262144*x^11 - 10321
92*x^9 + 746496*x^7 + 1166400*x^5 - 1364688*x^3 + 177147*x)*sqrt(4*x^2 - 9))/(20
48*x^6 - 6912*x^4 + 5832*x^2 - 4*(256*x^5 - 576*x^3 + 243*x)*sqrt(4*x^2 - 9) - 7
29)

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Sympy [A]  time = 15.5596, size = 167, normalized size = 2.32 \[ \begin{cases} \frac{2 x^{7}}{3 \sqrt{4 x^{2} - 9}} - \frac{15 x^{5}}{8 \sqrt{4 x^{2} - 9}} - \frac{27 x^{3}}{64 \sqrt{4 x^{2} - 9}} + \frac{729 x}{256 \sqrt{4 x^{2} - 9}} - \frac{729 \operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{512} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\- \frac{2 i x^{7}}{3 \sqrt{- 4 x^{2} + 9}} + \frac{15 i x^{5}}{8 \sqrt{- 4 x^{2} + 9}} + \frac{27 i x^{3}}{64 \sqrt{- 4 x^{2} + 9}} - \frac{729 i x}{256 \sqrt{- 4 x^{2} + 9}} + \frac{729 i \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{512} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*x**7/(3*sqrt(4*x**2 - 9)) - 15*x**5/(8*sqrt(4*x**2 - 9)) - 27*x**3/
(64*sqrt(4*x**2 - 9)) + 729*x/(256*sqrt(4*x**2 - 9)) - 729*acosh(2*x/3)/512, 4*A
bs(x**2)/9 > 1), (-2*I*x**7/(3*sqrt(-4*x**2 + 9)) + 15*I*x**5/(8*sqrt(-4*x**2 +
9)) + 27*I*x**3/(64*sqrt(-4*x**2 + 9)) - 729*I*x/(256*sqrt(-4*x**2 + 9)) + 729*I
*asin(2*x/3)/512, True))

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GIAC/XCAS [A]  time = 0.205984, size = 59, normalized size = 0.82 \[ \frac{1}{768} \,{\left (8 \,{\left (16 \, x^{2} - 9\right )} x^{2} - 243\right )} \sqrt{4 \, x^{2} - 9} x + \frac{729}{512} \,{\rm ln}\left ({\left | -2 \, x + \sqrt{4 \, x^{2} - 9} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/768*(8*(16*x^2 - 9)*x^2 - 243)*sqrt(4*x^2 - 9)*x + 729/512*ln(abs(-2*x + sqrt(
4*x^2 - 9)))