3.1414 \(\int \frac{1}{x^{13} (2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ \frac{5}{96 x^6 \sqrt{x^6+2}}-\frac{1}{24 x^{12} \sqrt{x^6+2}}+\frac{5}{64 \sqrt{x^6+2}}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{64 \sqrt{2}} \]

[Out]

5/(64*Sqrt[2 + x^6]) - 1/(24*x^12*Sqrt[2 + x^6]) + 5/(96*x^6*Sqrt[2 + x^6]) - (5*ArcTanh[Sqrt[2 + x^6]/Sqrt[2]
])/(64*Sqrt[2])

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Rubi [A]  time = 0.0303197, antiderivative size = 74, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 207} \[ \frac{5 \sqrt{x^6+2}}{64 x^6}-\frac{5 \sqrt{x^6+2}}{48 x^{12}}+\frac{1}{6 x^{12} \sqrt{x^6+2}}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{64 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^13*(2 + x^6)^(3/2)),x]

[Out]

1/(6*x^12*Sqrt[2 + x^6]) - (5*Sqrt[2 + x^6])/(48*x^12) + (5*Sqrt[2 + x^6])/(64*x^6) - (5*ArcTanh[Sqrt[2 + x^6]
/Sqrt[2]])/(64*Sqrt[2])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^{13} \left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x^3 (2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac{1}{6 x^{12} \sqrt{2+x^6}}+\frac{5}{12} \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{2+x}} \, dx,x,x^6\right )\\ &=\frac{1}{6 x^{12} \sqrt{2+x^6}}-\frac{5 \sqrt{2+x^6}}{48 x^{12}}-\frac{5}{32} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{2+x}} \, dx,x,x^6\right )\\ &=\frac{1}{6 x^{12} \sqrt{2+x^6}}-\frac{5 \sqrt{2+x^6}}{48 x^{12}}+\frac{5 \sqrt{2+x^6}}{64 x^6}+\frac{5}{128} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{2+x}} \, dx,x,x^6\right )\\ &=\frac{1}{6 x^{12} \sqrt{2+x^6}}-\frac{5 \sqrt{2+x^6}}{48 x^{12}}+\frac{5 \sqrt{2+x^6}}{64 x^6}+\frac{5}{64} \operatorname{Subst}\left (\int \frac{1}{-2+x^2} \, dx,x,\sqrt{2+x^6}\right )\\ &=\frac{1}{6 x^{12} \sqrt{2+x^6}}-\frac{5 \sqrt{2+x^6}}{48 x^{12}}+\frac{5 \sqrt{2+x^6}}{64 x^6}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{2+x^6}}{\sqrt{2}}\right )}{64 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0057099, size = 30, normalized size = 0.42 \[ \frac{\, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{x^6}{2}+1\right )}{24 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^13*(2 + x^6)^(3/2)),x]

[Out]

Hypergeometric2F1[-1/2, 3, 1/2, 1 + x^6/2]/(24*Sqrt[2 + x^6])

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Maple [A]  time = 0.022, size = 51, normalized size = 0.7 \begin{align*}{\frac{15\,{x}^{12}+10\,{x}^{6}-8}{192\,{x}^{12}}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{5\,\sqrt{2}}{128}\ln \left ({ \left ( \sqrt{{x}^{6}+2}-\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{6}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^13/(x^6+2)^(3/2),x)

[Out]

1/192*(15*x^12+10*x^6-8)/x^12/(x^6+2)^(1/2)+5/128*2^(1/2)*ln(((x^6+2)^(1/2)-2^(1/2))/(x^6)^(1/2))

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Maxima [A]  time = 1.49913, size = 109, normalized size = 1.54 \begin{align*} \frac{5}{256} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) - \frac{50 \, x^{6} - 15 \,{\left (x^{6} + 2\right )}^{2} + 68}{192 \,{\left ({\left (x^{6} + 2\right )}^{\frac{5}{2}} - 4 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} + 4 \, \sqrt{x^{6} + 2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

5/256*sqrt(2)*log(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) - 1/192*(50*x^6 - 15*(x^6 + 2)^2 + 68)
/((x^6 + 2)^(5/2) - 4*(x^6 + 2)^(3/2) + 4*sqrt(x^6 + 2))

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Fricas [A]  time = 1.48157, size = 186, normalized size = 2.62 \begin{align*} \frac{15 \, \sqrt{2}{\left (x^{18} + 2 \, x^{12}\right )} \log \left (\frac{x^{6} - 2 \, \sqrt{2} \sqrt{x^{6} + 2} + 4}{x^{6}}\right ) + 4 \,{\left (15 \, x^{12} + 10 \, x^{6} - 8\right )} \sqrt{x^{6} + 2}}{768 \,{\left (x^{18} + 2 \, x^{12}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

1/768*(15*sqrt(2)*(x^18 + 2*x^12)*log((x^6 - 2*sqrt(2)*sqrt(x^6 + 2) + 4)/x^6) + 4*(15*x^12 + 10*x^6 - 8)*sqrt
(x^6 + 2))/(x^18 + 2*x^12)

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Sympy [A]  time = 6.30415, size = 68, normalized size = 0.96 \begin{align*} - \frac{5 \sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{128} + \frac{5}{64 x^{3} \sqrt{1 + \frac{2}{x^{6}}}} + \frac{5}{96 x^{9} \sqrt{1 + \frac{2}{x^{6}}}} - \frac{1}{24 x^{15} \sqrt{1 + \frac{2}{x^{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**13/(x**6+2)**(3/2),x)

[Out]

-5*sqrt(2)*asinh(sqrt(2)/x**3)/128 + 5/(64*x**3*sqrt(1 + 2/x**6)) + 5/(96*x**9*sqrt(1 + 2/x**6)) - 1/(24*x**15
*sqrt(1 + 2/x**6))

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Giac [A]  time = 1.16682, size = 92, normalized size = 1.3 \begin{align*} \frac{5}{256} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) + \frac{1}{24 \, \sqrt{x^{6} + 2}} + \frac{7 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - 18 \, \sqrt{x^{6} + 2}}{192 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

5/256*sqrt(2)*log(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) + 1/24/sqrt(x^6 + 2) + 1/192*(7*(x^6 +
2)^(3/2) - 18*sqrt(x^6 + 2))/x^12