3.14.90 \(\int \frac {x^{14}}{\sqrt {2+x^6}} \, dx\) [1390]

Optimal. Leaf size=47 \[ -\frac {1}{4} x^3 \sqrt {2+x^6}+\frac {1}{12} x^9 \sqrt {2+x^6}+\frac {1}{2} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right ) \]

[Out]

1/2*arcsinh(1/2*x^3*2^(1/2))-1/4*x^3*(x^6+2)^(1/2)+1/12*x^9*(x^6+2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 327, 221} \begin {gather*} \frac {1}{2} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )+\frac {1}{12} \sqrt {x^6+2} x^9-\frac {1}{4} \sqrt {x^6+2} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^14/Sqrt[2 + x^6],x]

[Out]

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

Rule 221

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 281

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 327

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^{14}}{\sqrt {2+x^6}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^4}{\sqrt {2+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{12} x^9 \sqrt {2+x^6}-\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{4} x^3 \sqrt {2+x^6}+\frac {1}{12} x^9 \sqrt {2+x^6}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{4} x^3 \sqrt {2+x^6}+\frac {1}{12} x^9 \sqrt {2+x^6}+\frac {1}{2} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 40, normalized size = 0.85 \begin {gather*} \frac {1}{12} x^3 \left (-3+x^6\right ) \sqrt {2+x^6}+\frac {1}{2} \tanh ^{-1}\left (\frac {x^3}{\sqrt {2+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^14/Sqrt[2 + x^6],x]

[Out]

(x^3*(-3 + x^6)*Sqrt[2 + x^6])/12 + ArcTanh[x^3/Sqrt[2 + x^6]]/2

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Maple [A]
time = 0.18, size = 30, normalized size = 0.64

method	result	size
risch	\(\frac {x^{3} \left (x^{6}-3\right ) \sqrt {x^{6}+2}}{12}+\frac {\arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )}{2}\)	\(30\)
trager	\(\frac {x^{3} \left (x^{6}-3\right ) \sqrt {x^{6}+2}}{12}+\frac {\ln \left (x^{3}+\sqrt {x^{6}+2}\right )}{2}\)	\(33\)
meijerg	\(\frac {-\frac {\sqrt {\pi }\, x^{3} \sqrt {2}\, \left (-5 x^{6}+15\right ) \sqrt {1+\frac {x^{6}}{2}}}{60}+\frac {\sqrt {\pi }\, \arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )}{2}}{\sqrt {\pi }}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(x^6+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/12*x^3*(x^6-3)*(x^6+2)^(1/2)+1/2*arcsinh(1/2*x^3*2^(1/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (36) = 72\).
time = 0.29, size = 86, normalized size = 1.83 \begin {gather*} -\frac {\frac {5 \, \sqrt {x^{6} + 2}}{x^{3}} - \frac {3 \, {\left (x^{6} + 2\right )}^{\frac {3}{2}}}{x^{9}}}{6 \, {\left (\frac {2 \, {\left (x^{6} + 2\right )}}{x^{6}} - \frac {{\left (x^{6} + 2\right )}^{2}}{x^{12}} - 1\right )}} + \frac {1}{4} \, \log \left (\frac {\sqrt {x^{6} + 2}}{x^{3}} + 1\right ) - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{6} + 2}}{x^{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

-1/6*(5*sqrt(x^6 + 2)/x^3 - 3*(x^6 + 2)^(3/2)/x^9)/(2*(x^6 + 2)/x^6 - (x^6 + 2)^2/x^12 - 1) + 1/4*log(sqrt(x^6
+ 2)/x^3 + 1) - 1/4*log(sqrt(x^6 + 2)/x^3 - 1)

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Fricas [A]
time = 0.35, size = 35, normalized size = 0.74 \begin {gather*} \frac {1}{12} \, {\left (x^{9} - 3 \, x^{3}\right )} \sqrt {x^{6} + 2} - \frac {1}{2} \, \log \left (-x^{3} + \sqrt {x^{6} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

1/12*(x^9 - 3*x^3)*sqrt(x^6 + 2) - 1/2*log(-x^3 + sqrt(x^6 + 2))

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Sympy [A]
time = 2.37, size = 53, normalized size = 1.13 \begin {gather*} \frac {x^{15}}{12 \sqrt {x^{6} + 2}} - \frac {x^{9}}{12 \sqrt {x^{6} + 2}} - \frac {x^{3}}{2 \sqrt {x^{6} + 2}} + \frac {\operatorname {asinh}{\left (\frac {\sqrt {2} x^{3}}{2} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14/(x**6+2)**(1/2),x)

[Out]

x**15/(12*sqrt(x**6 + 2)) - x**9/(12*sqrt(x**6 + 2)) - x**3/(2*sqrt(x**6 + 2)) + asinh(sqrt(2)*x**3/2)/2

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

integrate(x^14/sqrt(x^6 + 2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{14}}{\sqrt {x^6+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(x^6 + 2)^(1/2),x)

[Out]

int(x^14/(x^6 + 2)^(1/2), x)

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