3.1390 \(\int \frac{x^{14}}{\sqrt{2+x^6}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0197368, antiderivative size = 47, normalized size of antiderivative = 1., 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 = {275, 321, 215} \[ \frac{1}{12} \sqrt{x^6+2} x^9-\frac{1}{4} \sqrt{x^6+2} x^3+\frac{1}{2} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^14/Sqrt[2 + x^6],x]

[Out]

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

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]

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]

Rubi steps

\begin{align*} \int \frac{x^{14}}{\sqrt{2+x^6}} \, dx &=\frac{1}{3} \operatorname{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} \operatorname{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} \operatorname{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*}

Mathematica [A]  time = 0.0105975, size = 35, normalized size = 0.74 \[ \frac{1}{12} \left (\left (x^6-3\right ) \sqrt{x^6+2} x^3+6 \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^14/Sqrt[2 + x^6],x]

[Out]

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

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Maple [A]  time = 0.029, size = 30, normalized size = 0.6 \begin{align*}{\frac{{x}^{3} \left ({x}^{6}-3 \right ) }{12}\sqrt{{x}^{6}+2}}+{\frac{1}{2}{\it Arcsinh} \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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]  time = 0.995798, size = 116, normalized size = 2.47 \begin{align*} -\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{align*}

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 = 1.47271, size = 89, normalized size = 1.89 \begin{align*} \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{align*}

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 = 3.92494, size = 53, normalized size = 1.13 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{14}}{\sqrt{x^{6} + 2}}\,{d x} \end{align*}

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)