3.7.76 \(\int x^{20} \sqrt {-1+x^6} \, dx\)

Optimal. Leaf size=53 \[ \frac {\sqrt {x^6-1} \left (48 x^{21}-8 x^{15}-10 x^9-15 x^3\right )}{1152}-\frac {5}{384} \log \left (\sqrt {x^6-1}+x^3\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.57, number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {275, 279, 321, 217, 206} \begin {gather*} \frac {1}{24} \sqrt {x^6-1} x^{21}-\frac {1}{144} \sqrt {x^6-1} x^{15}-\frac {5}{576} \sqrt {x^6-1} x^9-\frac {5}{384} \sqrt {x^6-1} x^3-\frac {5}{384} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^20*Sqrt[-1 + x^6],x]

[Out]

(-5*x^3*Sqrt[-1 + x^6])/384 - (5*x^9*Sqrt[-1 + x^6])/576 - (x^15*Sqrt[-1 + x^6])/144 + (x^21*Sqrt[-1 + x^6])/2
4 - (5*ArcTanh[x^3/Sqrt[-1 + x^6]])/384

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])

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 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]

Rubi steps

\begin {align*} \int x^{20} \sqrt {-1+x^6} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^6 \sqrt {-1+x^2} \, dx,x,x^3\right )\\ &=\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{144} x^{15} \sqrt {-1+x^6}+\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {5}{144} \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {5}{576} x^9 \sqrt {-1+x^6}-\frac {1}{144} x^{15} \sqrt {-1+x^6}+\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {5}{192} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {5}{384} x^3 \sqrt {-1+x^6}-\frac {5}{576} x^9 \sqrt {-1+x^6}-\frac {1}{144} x^{15} \sqrt {-1+x^6}+\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {5}{384} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {5}{384} x^3 \sqrt {-1+x^6}-\frac {5}{576} x^9 \sqrt {-1+x^6}-\frac {1}{144} x^{15} \sqrt {-1+x^6}+\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {5}{384} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=-\frac {5}{384} x^3 \sqrt {-1+x^6}-\frac {5}{576} x^9 \sqrt {-1+x^6}-\frac {1}{144} x^{15} \sqrt {-1+x^6}+\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {5}{384} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 61, normalized size = 1.15 \begin {gather*} \frac {\left (x^6-1\right ) \left (15 \sin ^{-1}\left (x^3\right )+\sqrt {1-x^6} \left (48 x^{18}-8 x^{12}-10 x^6-15\right ) x^3\right )}{1152 \sqrt {-\left (x^6-1\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^20*Sqrt[-1 + x^6],x]

[Out]

((-1 + x^6)*(x^3*Sqrt[1 - x^6]*(-15 - 10*x^6 - 8*x^12 + 48*x^18) + 15*ArcSin[x^3]))/(1152*Sqrt[-(-1 + x^6)^2])

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IntegrateAlgebraic [A]  time = 0.15, size = 53, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (-15 x^3-10 x^9-8 x^{15}+48 x^{21}\right )}{1152}-\frac {5}{384} \log \left (x^3+\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^20*Sqrt[-1 + x^6],x]

[Out]

(Sqrt[-1 + x^6]*(-15*x^3 - 10*x^9 - 8*x^15 + 48*x^21))/1152 - (5*Log[x^3 + Sqrt[-1 + x^6]])/384

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fricas [A]  time = 0.58, size = 47, normalized size = 0.89 \begin {gather*} \frac {1}{1152} \, {\left (48 \, x^{21} - 8 \, x^{15} - 10 \, x^{9} - 15 \, x^{3}\right )} \sqrt {x^{6} - 1} + \frac {5}{384} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^20*(x^6-1)^(1/2),x, algorithm="fricas")

[Out]

1/1152*(48*x^21 - 8*x^15 - 10*x^9 - 15*x^3)*sqrt(x^6 - 1) + 5/384*log(-x^3 + sqrt(x^6 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{6} - 1} x^{20}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^20*(x^6-1)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(x^6 - 1)*x^20, x)

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maple [A]  time = 0.23, size = 47, normalized size = 0.89

method	result	size
trager	\(\frac {x^{3} \left (48 x^{18}-8 x^{12}-10 x^{6}-15\right ) \sqrt {x^{6}-1}}{1152}+\frac {5 \ln \left (x^{3}-\sqrt {x^{6}-1}\right )}{384}\)	\(47\)
risch	\(\frac {x^{3} \left (48 x^{18}-8 x^{12}-10 x^{6}-15\right ) \sqrt {x^{6}-1}}{1152}-\frac {5 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{384 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\)	\(55\)
meijerg	\(-\frac {i \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {i \sqrt {\pi }\, x^{3} \left (-336 x^{18}+56 x^{12}+70 x^{6}+105\right ) \sqrt {-x^{6}+1}}{672}+\frac {5 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )}{32}\right )}{12 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^20*(x^6-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/1152*x^3*(48*x^18-8*x^12-10*x^6-15)*(x^6-1)^(1/2)+5/384*ln(x^3-(x^6-1)^(1/2))

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maxima [B]  time = 0.31, size = 134, normalized size = 2.53 \begin {gather*} -\frac {\frac {15 \, \sqrt {x^{6} - 1}}{x^{3}} + \frac {73 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{x^{9}} - \frac {55 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}}}{x^{15}} + \frac {15 \, {\left (x^{6} - 1\right )}^{\frac {7}{2}}}{x^{21}}}{1152 \, {\left (\frac {4 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {6 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {4 \, {\left (x^{6} - 1\right )}^{3}}{x^{18}} - \frac {{\left (x^{6} - 1\right )}^{4}}{x^{24}} - 1\right )}} - \frac {5}{768} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) + \frac {5}{768} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^20*(x^6-1)^(1/2),x, algorithm="maxima")

[Out]

-1/1152*(15*sqrt(x^6 - 1)/x^3 + 73*(x^6 - 1)^(3/2)/x^9 - 55*(x^6 - 1)^(5/2)/x^15 + 15*(x^6 - 1)^(7/2)/x^21)/(4
*(x^6 - 1)/x^6 - 6*(x^6 - 1)^2/x^12 + 4*(x^6 - 1)^3/x^18 - (x^6 - 1)^4/x^24 - 1) - 5/768*log(sqrt(x^6 - 1)/x^3
+ 1) + 5/768*log(sqrt(x^6 - 1)/x^3 - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^{20}\,\sqrt {x^6-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^20*(x^6 - 1)^(1/2),x)

[Out]

int(x^20*(x^6 - 1)^(1/2), x)

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sympy [A]  time = 6.23, size = 175, normalized size = 3.30 \begin {gather*} \begin {cases} \frac {x^{27}}{24 \sqrt {x^{6} - 1}} - \frac {7 x^{21}}{144 \sqrt {x^{6} - 1}} - \frac {x^{15}}{576 \sqrt {x^{6} - 1}} - \frac {5 x^{9}}{1152 \sqrt {x^{6} - 1}} + \frac {5 x^{3}}{384 \sqrt {x^{6} - 1}} - \frac {5 \operatorname {acosh}{\left (x^{3} \right )}}{384} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{27}}{24 \sqrt {1 - x^{6}}} + \frac {7 i x^{21}}{144 \sqrt {1 - x^{6}}} + \frac {i x^{15}}{576 \sqrt {1 - x^{6}}} + \frac {5 i x^{9}}{1152 \sqrt {1 - x^{6}}} - \frac {5 i x^{3}}{384 \sqrt {1 - x^{6}}} + \frac {5 i \operatorname {asin}{\left (x^{3} \right )}}{384} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**20*(x**6-1)**(1/2),x)

[Out]

Piecewise((x**27/(24*sqrt(x**6 - 1)) - 7*x**21/(144*sqrt(x**6 - 1)) - x**15/(576*sqrt(x**6 - 1)) - 5*x**9/(115
2*sqrt(x**6 - 1)) + 5*x**3/(384*sqrt(x**6 - 1)) - 5*acosh(x**3)/384, Abs(x**6) > 1), (-I*x**27/(24*sqrt(1 - x*
*6)) + 7*I*x**21/(144*sqrt(1 - x**6)) + I*x**15/(576*sqrt(1 - x**6)) + 5*I*x**9/(1152*sqrt(1 - x**6)) - 5*I*x*
*3/(384*sqrt(1 - x**6)) + 5*I*asin(x**3)/384, True))

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