3.1416 \(\int \frac{x^8}{(2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )-\frac{x^3}{3 \sqrt{x^6+2}} \]

[Out]

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

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Rubi [A]  time = 0.012157, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 288, 215} \[ \frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )-\frac{x^3}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(2 + x^6)^(3/2),x]

[Out]

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

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 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 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^8}{\left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\left (2+x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=-\frac{x^3}{3 \sqrt{2+x^6}}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+x^2}} \, dx,x,x^3\right )\\ &=-\frac{x^3}{3 \sqrt{2+x^6}}+\frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0148155, size = 29, normalized size = 0.94 \[ \frac{1}{3} \left (\sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )-\frac{x^3}{\sqrt{x^6+2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(2 + x^6)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.012, size = 25, normalized size = 0.8 \begin{align*}{\frac{1}{3}{\it Arcsinh} \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ) }-{\frac{{x}^{3}}{3}{\frac{1}{\sqrt{{x}^{6}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^6+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.991835, size = 61, normalized size = 1.97 \begin{align*} -\frac{x^{3}}{3 \, \sqrt{x^{6} + 2}} + \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} + 1\right ) - \frac{1}{6} \, \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^8/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45144, size = 113, normalized size = 3.65 \begin{align*} -\frac{x^{6} + \sqrt{x^{6} + 2} x^{3} +{\left (x^{6} + 2\right )} \log \left (-x^{3} + \sqrt{x^{6} + 2}\right ) + 2}{3 \,{\left (x^{6} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.57409, size = 26, normalized size = 0.84 \begin{align*} - \frac{x^{3}}{3 \sqrt{x^{6} + 2}} + \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(x**6+2)**(3/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^8/(x^6 + 2)^(3/2), x)