3.15.16 \(\int \frac {x^8}{(2+x^6)^{3/2}} \, dx\) [1416]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 = {281, 294, 221} \begin {gather*} \frac {1}{3} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )-\frac {x^3}{3 \sqrt {x^6+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/(2 + x^6)^(3/2),x]

[Out]

-1/3*x^3/Sqrt[2 + x^6] + ArcSinh[x^3/Sqrt[2]]/3

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 294

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]

Rubi steps

\begin {align*} \int \frac {x^8}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{3} \text {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} \text {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*}

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Mathematica [A]
time = 0.13, size = 35, normalized size = 1.13 \begin {gather*} -\frac {x^3}{3 \sqrt {2+x^6}}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {2+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/(2 + x^6)^(3/2),x]

[Out]

-1/3*x^3/Sqrt[2 + x^6] + ArcTanh[x^3/Sqrt[2 + x^6]]/3

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Maple [A]
time = 0.18, size = 25, normalized size = 0.81

method	result	size
risch	\(\frac {\arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )}{3}-\frac {x^{3}}{3 \sqrt {x^{6}+2}}\)	\(25\)
trager	\(-\frac {x^{3}}{3 \sqrt {x^{6}+2}}-\frac {\ln \left (x^{3}-\sqrt {x^{6}+2}\right )}{3}\)	\(30\)
meijerg	\(\frac {-\frac {\sqrt {\pi }\, x^{3} \sqrt {2}}{2 \sqrt {1+\frac {x^{6}}{2}}}+\sqrt {\pi }\, \arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )}{3 \sqrt {\pi }}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.30, size = 45, normalized size = 1.45 \begin {gather*} -\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 {gather*}

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 = 0.37, size = 45, normalized size = 1.45 \begin {gather*} -\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 {gather*}

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 = 0.72, size = 26, normalized size = 0.84 \begin {gather*} - \frac {x^{3}}{3 \sqrt {x^{6} + 2}} + \frac {\operatorname {asinh}{\left (\frac {\sqrt {2} x^{3}}{2} \right )}}{3} \end {gather*}

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 = 0.00, 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^8/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^8/(x^6 + 2)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^8}{{\left (x^6+2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^6 + 2)^(3/2),x)

[Out]

int(x^8/(x^6 + 2)^(3/2), x)

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