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

Optimal. Leaf size=63 \[ -\frac{81}{256} \sqrt{4 x^2+9} x+\frac{1}{6} \sqrt{4 x^2+9} x^5+\frac{3}{32} \sqrt{4 x^2+9} x^3+\frac{729}{512} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

[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*ArcSinh[(2*x)/3])/512

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Rubi [A]  time = 0.0590177, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{81}{256} \sqrt{4 x^2+9} x+\frac{1}{6} \sqrt{4 x^2+9} x^5+\frac{3}{32} \sqrt{4 x^2+9} x^3+\frac{729}{512} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

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*ArcSinh[(2*x)/3])/512

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Rubi in Sympy [A]  time = 7.55554, size = 56, normalized size = 0.89 \[ \frac{x^{5} \sqrt{4 x^{2} + 9}}{6} + \frac{3 x^{3} \sqrt{4 x^{2} + 9}}{32} - \frac{81 x \sqrt{4 x^{2} + 9}}{256} + \frac{729 \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{512} \]

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*asinh(2*x/3)/512

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Mathematica [A]  time = 0.022538, size = 39, normalized size = 0.62 \[ \frac{1}{768} x \sqrt{4 x^2+9} \left (128 x^4+72 x^2-243\right )+\frac{729}{512} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

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*ArcSinh[(2*x)/3])/512

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Maple [A]  time = 0.008, size = 46, normalized size = 0.7 \[{\frac{{x}^{3}}{24} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{9\,x}{128} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}+{\frac{81\,x}{256}\sqrt{4\,{x}^{2}+9}}+{\frac{729}{512}{\it Arcsinh} \left ({\frac{2\,x}{3}} \right ) } \]

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)+729/51
2*arcsinh(2/3*x)

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Maxima [A]  time = 1.4954, size = 61, normalized size = 0.97 \[ \frac{1}{24} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} x^{3} - \frac{9}{128} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} x + \frac{81}{256} \, \sqrt{4 \, x^{2} + 9} x + \frac{729}{512} \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \]

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*arcsinh(2/3*x)

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Fricas [A]  time = 0.227408, size = 236, normalized size = 3.75 \[ -\frac{1048576 \, x^{12} + 5308416 \, x^{10} + 6967296 \, x^{8} - 3172608 \, x^{6} - 10707552 \, x^{4} - 4251528 \, x^{2} + 2187 \,{\left (2048 \, x^{6} + 6912 \, x^{4} + 5832 \, x^{2} - 4 \,{\left (256 \, x^{5} + 576 \, x^{3} + 243 \, x\right )} \sqrt{4 \, x^{2} + 9} + 729\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) - 2 \,{\left (262144 \, x^{11} + 1032192 \, x^{9} + 746496 \, x^{7} - 1166400 \, x^{5} - 1364688 \, x^{3} - 177147 \, x\right )} \sqrt{4 \, x^{2} + 9}}{1536 \,{\left (2048 \, x^{6} + 6912 \, x^{4} + 5832 \, x^{2} - 4 \,{\left (256 \, x^{5} + 576 \, x^{3} + 243 \, x\right )} \sqrt{4 \, x^{2} + 9} + 729\right )}} \]

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 = 15.3063, size = 75, normalized size = 1.19 \[ \frac{2 x^{7}}{3 \sqrt{4 x^{2} + 9}} + \frac{15 x^{5}}{8 \sqrt{4 x^{2} + 9}} - \frac{27 x^{3}}{64 \sqrt{4 x^{2} + 9}} - \frac{729 x}{256 \sqrt{4 x^{2} + 9}} + \frac{729 \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{512} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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*asinh(2*x/3)/512

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GIAC/XCAS [A]  time = 0.206247, size = 58, normalized size = 0.92 \[ \frac{1}{768} \,{\left (8 \,{\left (16 \, x^{2} + 9\right )} x^{2} - 243\right )} \sqrt{4 \, x^{2} + 9} x - \frac{729}{512} \,{\rm ln}\left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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