3.1380 \(\int x^8 \sqrt{-1+4 x^6} \, dx\)

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 = 0.0290294, 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, Rules used = {275, 279, 321, 217, 206} \[ \frac{1}{12} \sqrt{4 x^6-1} x^9-\frac{1}{96} \sqrt{4 x^6-1} x^3-\frac{1}{192} \tanh ^{-1}\left (\frac{2 x^3}{\sqrt{4 x^6-1}}\right ) \]

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^8 \sqrt{-1+4 x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 \sqrt{-1+4 x^2} \, dx,x,x^3\right )\\ &=\frac{1}{12} x^9 \sqrt{-1+4 x^6}-\frac{1}{12} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-1+4 x^2}} \, dx,x,x^3\right )\\ &=-\frac{1}{96} x^3 \sqrt{-1+4 x^6}+\frac{1}{12} x^9 \sqrt{-1+4 x^6}-\frac{1}{96} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+4 x^2}} \, dx,x,x^3\right )\\ &=-\frac{1}{96} x^3 \sqrt{-1+4 x^6}+\frac{1}{12} x^9 \sqrt{-1+4 x^6}-\frac{1}{96} \operatorname{Subst}\left (\int \frac{1}{1-4 x^2} \, dx,x,\frac{x^3}{\sqrt{-1+4 x^6}}\right )\\ &=-\frac{1}{96} x^3 \sqrt{-1+4 x^6}+\frac{1}{12} x^9 \sqrt{-1+4 x^6}-\frac{1}{192} \tanh ^{-1}\left (\frac{2 x^3}{\sqrt{-1+4 x^6}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0292718, size = 56, normalized size = 0.97 \[ \frac{\left (4 x^6-1\right ) \left (2 \sqrt{1-4 x^6} \left (8 x^6-1\right ) x^3+\sin ^{-1}\left (2 x^3\right )\right )}{192 \sqrt{-\left (1-4 x^6\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*Sqrt[-1 + 4*x^6],x]

[Out]

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

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Maple [C]  time = 0.049, size = 53, normalized size = 0.9 \begin{align*}{\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 ) }}}} \end{align*}

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 [B]  time = 0.990004, size = 131, normalized size = 2.26 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(4*x^6-1)^(1/2),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*lo
g(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 = 1.47541, size = 100, normalized size = 1.72 \begin{align*} \frac{1}{96} \,{\left (8 \, x^{9} - x^{3}\right )} \sqrt{4 \, x^{6} - 1} + \frac{1}{192} \, \log \left (-2 \, x^{3} + \sqrt{4 \, x^{6} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(4*x^6-1)^(1/2),x, algorithm="fricas")

[Out]

1/96*(8*x^9 - x^3)*sqrt(4*x^6 - 1) + 1/192*log(-2*x^3 + sqrt(4*x^6 - 1))

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Sympy [A]  time = 2.81705, size = 119, normalized size = 2.05 \begin{align*} \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{1 - 4 x^{6}}} + \frac{i x^{9}}{8 \sqrt{1 - 4 x^{6}}} - \frac{i x^{3}}{96 \sqrt{1 - 4 x^{6}}} + \frac{i \operatorname{asin}{\left (2 x^{3} \right )}}{192} & \text{otherwise} \end{cases} \end{align*}

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, x^{6} - 1} x^{8}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(4*x^6-1)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(4*x^6 - 1)*x^8, x)