3.1378 \(\int x^8 \sqrt{-1+4 x^6} \, dx\)

Optimal. Leaf size=58 \[ \frac{1}{12} \sqrt{4 x^6-1} x^9-\frac{1}{96} \sqrt{4 x^6-1} x^3-\frac{1}{192} \tanh ^{-1}\left (\frac{2 x^3}{\sqrt{4 x^6-1}}\right ) \]

[Out]

-(x^3*Sqrt[-1 + 4*x^6])/96 + (x^9*Sqrt[-1 + 4*x^6])/12 - ArcTanh[(2*x^3)/Sqrt[-1
+ 4*x^6]]/192

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Rubi [A]  time = 0.0734214, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{12} \sqrt{4 x^6-1} x^9-\frac{1}{96} \sqrt{4 x^6-1} x^3-\frac{1}{192} \tanh ^{-1}\left (\frac{2 x^3}{\sqrt{4 x^6-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^8*Sqrt[-1 + 4*x^6],x]

[Out]

-(x^3*Sqrt[-1 + 4*x^6])/96 + (x^9*Sqrt[-1 + 4*x^6])/12 - ArcTanh[(2*x^3)/Sqrt[-1
+ 4*x^6]]/192

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Rubi in Sympy [A]  time = 8.12988, size = 48, normalized size = 0.83 \[ \frac{x^{9} \sqrt{4 x^{6} - 1}}{12} - \frac{x^{3} \sqrt{4 x^{6} - 1}}{96} - \frac{\operatorname{atanh}{\left (\frac{2 x^{3}}{\sqrt{4 x^{6} - 1}} \right )}}{192} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(4*x**6-1)**(1/2),x)

[Out]

x**9*sqrt(4*x**6 - 1)/12 - x**3*sqrt(4*x**6 - 1)/96 - atanh(2*x**3/sqrt(4*x**6 -
1))/192

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Mathematica [A]  time = 0.0323714, size = 48, normalized size = 0.83 \[ \frac{1}{192} \left (2 x^3 \sqrt{4 x^6-1} \left (8 x^6-1\right )-\log \left (\sqrt{4 x^6-1}+2 x^3\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*Sqrt[-1 + 4*x^6],x]

[Out]

(2*x^3*Sqrt[-1 + 4*x^6]*(-1 + 8*x^6) - Log[2*x^3 + Sqrt[-1 + 4*x^6]])/192

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Maple [C]  time = 0.074, size = 53, normalized size = 0.9 \[{\frac{{x}^{3} \left ( 8\,{x}^{6}-1 \right ) }{96}\sqrt{4\,{x}^{6}-1}}-{\frac{\arcsin \left ( 2\,{x}^{3} \right ) }{192}\sqrt{-{\it signum} \left ( 4\,{x}^{6}-1 \right ) }{\frac{1}{\sqrt{{\it signum} \left ( 4\,{x}^{6}-1 \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(4*x^6-1)^(1/2),x)

[Out]

1/96*x^3*(8*x^6-1)*(4*x^6-1)^(1/2)-1/192/signum(4*x^6-1)^(1/2)*(-signum(4*x^6-1)
)^(1/2)*arcsin(2*x^3)

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Maxima [A]  time = 1.43965, size = 131, normalized size = 2.26 \[ -\frac{\frac{4 \, \sqrt{4 \, x^{6} - 1}}{x^{3}} + \frac{{\left (4 \, x^{6} - 1\right )}^{\frac{3}{2}}}{x^{9}}}{96 \,{\left (\frac{8 \,{\left (4 \, x^{6} - 1\right )}}{x^{6}} - \frac{{\left (4 \, x^{6} - 1\right )}^{2}}{x^{12}} - 16\right )}} - \frac{1}{384} \, \log \left (\frac{\sqrt{4 \, x^{6} - 1}}{x^{3}} + 2\right ) + \frac{1}{384} \, \log \left (\frac{\sqrt{4 \, x^{6} - 1}}{x^{3}} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^6 - 1)*x^8,x, algorithm="maxima")

[Out]

-1/96*(4*sqrt(4*x^6 - 1)/x^3 + (4*x^6 - 1)^(3/2)/x^9)/(8*(4*x^6 - 1)/x^6 - (4*x^
6 - 1)^2/x^12 - 16) - 1/384*log(sqrt(4*x^6 - 1)/x^3 + 2) + 1/384*log(sqrt(4*x^6
- 1)/x^3 - 2)

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Fricas [A]  time = 0.227537, size = 193, normalized size = 3.33 \[ -\frac{4096 \, x^{24} - 2048 \, x^{18} + 320 \, x^{12} - 16 \, x^{6} -{\left (128 \, x^{12} - 32 \, x^{6} - 8 \,{\left (8 \, x^{9} - x^{3}\right )} \sqrt{4 \, x^{6} - 1} + 1\right )} \log \left (-2 \, x^{3} + \sqrt{4 \, x^{6} - 1}\right ) - 2 \,{\left (1024 \, x^{21} - 384 \, x^{15} + 40 \, x^{9} - x^{3}\right )} \sqrt{4 \, x^{6} - 1}}{192 \,{\left (128 \, x^{12} - 32 \, x^{6} - 8 \,{\left (8 \, x^{9} - x^{3}\right )} \sqrt{4 \, x^{6} - 1} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^6 - 1)*x^8,x, algorithm="fricas")

[Out]

-1/192*(4096*x^24 - 2048*x^18 + 320*x^12 - 16*x^6 - (128*x^12 - 32*x^6 - 8*(8*x^
9 - x^3)*sqrt(4*x^6 - 1) + 1)*log(-2*x^3 + sqrt(4*x^6 - 1)) - 2*(1024*x^21 - 384
*x^15 + 40*x^9 - x^3)*sqrt(4*x^6 - 1))/(128*x^12 - 32*x^6 - 8*(8*x^9 - x^3)*sqrt
(4*x^6 - 1) + 1)

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Sympy [A]  time = 10.082, size = 119, normalized size = 2.05 \[ \begin{cases} \frac{x^{15}}{3 \sqrt{4 x^{6} - 1}} - \frac{x^{9}}{8 \sqrt{4 x^{6} - 1}} + \frac{x^{3}}{96 \sqrt{4 x^{6} - 1}} - \frac{\operatorname{acosh}{\left (2 x^{3} \right )}}{192} & \text{for}\: 4 \left |{x^{6}}\right | > 1 \\- \frac{i x^{15}}{3 \sqrt{- 4 x^{6} + 1}} + \frac{i x^{9}}{8 \sqrt{- 4 x^{6} + 1}} - \frac{i x^{3}}{96 \sqrt{- 4 x^{6} + 1}} + \frac{i \operatorname{asin}{\left (2 x^{3} \right )}}{192} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(4*x**6-1)**(1/2),x)

[Out]

Piecewise((x**15/(3*sqrt(4*x**6 - 1)) - x**9/(8*sqrt(4*x**6 - 1)) + x**3/(96*sqr
t(4*x**6 - 1)) - acosh(2*x**3)/192, 4*Abs(x**6) > 1), (-I*x**15/(3*sqrt(-4*x**6
+ 1)) + I*x**9/(8*sqrt(-4*x**6 + 1)) - I*x**3/(96*sqrt(-4*x**6 + 1)) + I*asin(2*
x**3)/192, True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{4 \, x^{6} - 1} x^{8}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^6 - 1)*x^8,x, algorithm="giac")

[Out]

integrate(sqrt(4*x^6 - 1)*x^8, x)