3.7.15 \(\int \frac {x^{20}}{\sqrt {1+x^6}} \, dx\)

Optimal. Leaf size=48 \[ \frac {1}{144} \sqrt {x^6+1} \left (8 x^{15}-10 x^9+15 x^3\right )-\frac {5}{48} \log \left (\sqrt {x^6+1}+x^3\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 321, 215} \begin {gather*} -\frac {5}{48} \sinh ^{-1}\left (x^3\right )+\frac {1}{18} \sqrt {x^6+1} x^{15}-\frac {5}{72} \sqrt {x^6+1} x^9+\frac {5}{48} \sqrt {x^6+1} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^20/Sqrt[1 + x^6],x]

[Out]

(5*x^3*Sqrt[1 + x^6])/48 - (5*x^9*Sqrt[1 + x^6])/72 + (x^15*Sqrt[1 + x^6])/18 - (5*ArcSinh[x^3])/48

Rule 215

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

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 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 \frac {x^{20}}{\sqrt {1+x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{18} x^{15} \sqrt {1+x^6}-\frac {5}{18} \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {5}{72} x^9 \sqrt {1+x^6}+\frac {1}{18} x^{15} \sqrt {1+x^6}+\frac {5}{24} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=\frac {5}{48} x^3 \sqrt {1+x^6}-\frac {5}{72} x^9 \sqrt {1+x^6}+\frac {1}{18} x^{15} \sqrt {1+x^6}-\frac {5}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=\frac {5}{48} x^3 \sqrt {1+x^6}-\frac {5}{72} x^9 \sqrt {1+x^6}+\frac {1}{18} x^{15} \sqrt {1+x^6}-\frac {5}{48} \sinh ^{-1}\left (x^3\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 0.75 \begin {gather*} \frac {1}{144} \left (x^3 \sqrt {x^6+1} \left (8 x^{12}-10 x^6+15\right )-15 \sinh ^{-1}\left (x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^20/Sqrt[1 + x^6],x]

[Out]

(x^3*Sqrt[1 + x^6]*(15 - 10*x^6 + 8*x^12) - 15*ArcSinh[x^3])/144

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IntegrateAlgebraic [A]  time = 0.17, size = 48, normalized size = 1.00 \begin {gather*} \frac {1}{144} \sqrt {1+x^6} \left (15 x^3-10 x^9+8 x^{15}\right )-\frac {5}{48} \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))/144 - (5*Log[x^3 + Sqrt[1 + x^6]])/48

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fricas [A]  time = 0.45, size = 42, normalized size = 0.88 \begin {gather*} \frac {1}{144} \, {\left (8 \, x^{15} - 10 \, x^{9} + 15 \, x^{3}\right )} \sqrt {x^{6} + 1} + \frac {5}{48} \, \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]

1/144*(8*x^15 - 10*x^9 + 15*x^3)*sqrt(x^6 + 1) + 5/48*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 \frac {x^{20}}{\sqrt {x^{6} + 1}}\,{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(x^20/sqrt(x^6 + 1), x)

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maple [A]  time = 0.18, size = 32, normalized size = 0.67

method	result	size
risch	\(\frac {x^{3} \left (8 x^{12}-10 x^{6}+15\right ) \sqrt {x^{6}+1}}{144}-\frac {5 \arcsinh \left (x^{3}\right )}{48}\)	\(32\)
trager	\(\frac {x^{3} \left (8 x^{12}-10 x^{6}+15\right ) \sqrt {x^{6}+1}}{144}-\frac {5 \ln \left (x^{3}+\sqrt {x^{6}+1}\right )}{48}\)	\(40\)
meijerg	\(\frac {\frac {\sqrt {\pi }\, x^{3} \left (56 x^{12}-70 x^{6}+105\right ) \sqrt {x^{6}+1}}{168}-\frac {5 \sqrt {\pi }\, \arcsinh \left (x^{3}\right )}{8}}{6 \sqrt {\pi }}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*x^3*(8*x^12-10*x^6+15)*(x^6+1)^(1/2)-5/48*arcsinh(x^3)

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maxima [B]  time = 0.33, size = 109, normalized size = 2.27 \begin {gather*} \frac {\frac {33 \, \sqrt {x^{6} + 1}}{x^{3}} - \frac {40 \, {\left (x^{6} + 1\right )}^{\frac {3}{2}}}{x^{9}} + \frac {15 \, {\left (x^{6} + 1\right )}^{\frac {5}{2}}}{x^{15}}}{144 \, {\left (\frac {3 \, {\left (x^{6} + 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} + 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} + 1\right )}^{3}}{x^{18}} - 1\right )}} - \frac {5}{96} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) + \frac {5}{96} \, \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]

1/144*(33*sqrt(x^6 + 1)/x^3 - 40*(x^6 + 1)^(3/2)/x^9 + 15*(x^6 + 1)^(5/2)/x^15)/(3*(x^6 + 1)/x^6 - 3*(x^6 + 1)
^2/x^12 + (x^6 + 1)^3/x^18 - 1) - 5/96*log(sqrt(x^6 + 1)/x^3 + 1) + 5/96*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 \frac {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 = 4.72, size = 65, normalized size = 1.35 \begin {gather*} \frac {x^{21}}{18 \sqrt {x^{6} + 1}} - \frac {x^{15}}{72 \sqrt {x^{6} + 1}} + \frac {5 x^{9}}{144 \sqrt {x^{6} + 1}} + \frac {5 x^{3}}{48 \sqrt {x^{6} + 1}} - \frac {5 \operatorname {asinh}{\left (x^{3} \right )}}{48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**21/(18*sqrt(x**6 + 1)) - x**15/(72*sqrt(x**6 + 1)) + 5*x**9/(144*sqrt(x**6 + 1)) + 5*x**3/(48*sqrt(x**6 + 1
)) - 5*asinh(x**3)/48

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