3.1434 \(\int \frac {1}{x^2 (2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=408 \[ -\frac {\sqrt {x^6+2}}{3 x}+\frac {1}{6 \sqrt {x^6+2} x}+\frac {\left (1+\sqrt {3}\right ) \sqrt {x^6+2} x}{3 \left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )}-\frac {\left (1-\sqrt {3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt [3]{2} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3^{3/4} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}} \]

[Out]

1/6/x/(x^6+2)^(1/2)-1/3*(x^6+2)^(1/2)/x+1/3*x*(1+3^(1/2))*(x^6+2)^(1/2)/(2^(1/3)+x^2*(1+3^(1/2)))-1/3*3^(1/4)*
x*(2^(1/3)+x^2)*((2^(1/3)+x^2*(1-3^(1/2)))^2/(2^(1/3)+x^2*(1+3^(1/2)))^2)^(1/2)/(2^(1/3)+x^2*(1-3^(1/2)))*(2^(
1/3)+x^2*(1+3^(1/2)))*EllipticE((1-(2^(1/3)+x^2*(1-3^(1/2)))^2/(2^(1/3)+x^2*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+
1/4*2^(1/2))*((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+3^(1/2)))^2)^(1/2)*2^(1/3)/(x^6+2)^(1/2)/(x^2*(2^(1/3)
+x^2)/(2^(1/3)+x^2*(1+3^(1/2)))^2)^(1/2)-1/18*x*(2^(1/3)+x^2)*((2^(1/3)+x^2*(1-3^(1/2)))^2/(2^(1/3)+x^2*(1+3^(
1/2)))^2)^(1/2)/(2^(1/3)+x^2*(1-3^(1/2)))*(2^(1/3)+x^2*(1+3^(1/2)))*EllipticF((1-(2^(1/3)+x^2*(1-3^(1/2)))^2/(
2^(1/3)+x^2*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*(1-3^(1/2))*((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2
*(1+3^(1/2)))^2)^(1/2)*2^(1/3)*3^(3/4)/(x^6+2)^(1/2)/(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+3^(1/2)))^2)^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {290, 325, 308, 225, 1881} \[ \frac {\left (1+\sqrt {3}\right ) \sqrt {x^6+2} x}{3 \left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )}-\frac {\sqrt {x^6+2}}{3 x}+\frac {1}{6 \sqrt {x^6+2} x}-\frac {\left (1-\sqrt {3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt [3]{2} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3^{3/4} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(2 + x^6)^(3/2)),x]

[Out]

1/(6*x*Sqrt[2 + x^6]) - Sqrt[2 + x^6]/(3*x) + ((1 + Sqrt[3])*x*Sqrt[2 + x^6])/(3*(2^(1/3) + (1 + Sqrt[3])*x^2)
) - (2^(1/3)*x*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2 + x^4)/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*EllipticE[A
rcCos[(2^(1/3) + (1 - Sqrt[3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(3^(3/4)*Sqrt[(x^2*(2^(1
/3) + x^2))/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*Sqrt[2 + x^6]) - ((1 - Sqrt[3])*x*(2^(1/3) + x^2)*Sqrt[(2^(2/3) -
2^(1/3)*x^2 + x^4)/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*EllipticF[ArcCos[(2^(1/3) + (1 - Sqrt[3])*x^2)/(2^(1/3) +
(1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(3*2^(2/3)*3^(1/4)*Sqrt[(x^2*(2^(1/3) + x^2))/(2^(1/3) + (1 + Sqrt[3])
*x^2)^2]*Sqrt[2 + x^6])

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(x*(s
+ r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2
)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[(r*x^2*(s + r*x^2))/(s + (1
+ Sqrt[3])*r*x^2)^2]), x]] /; FreeQ[{a, b}, x]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 308

Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(
(Sqrt[3] - 1)*s^2)/(2*r^2), Int[1/Sqrt[a + b*x^6], x], x] - Dist[1/(2*r^2), Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4
)/Sqrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 1881

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/
a, 3]]}, Simp[((1 + Sqrt[3])*d*s^3*x*Sqrt[a + b*x^6])/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2)), x] - Simp[(3^(1/4)*
d*s*x*(s + r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticE[ArcCos[(s + (1 - Sqrt[
3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*
x^2)^2]*Sqrt[a + b*x^6]), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6 x \sqrt {2+x^6}}+\frac {2}{3} \int \frac {1}{x^2 \sqrt {2+x^6}} \, dx\\ &=\frac {1}{6 x \sqrt {2+x^6}}-\frac {\sqrt {2+x^6}}{3 x}+\frac {2}{3} \int \frac {x^4}{\sqrt {2+x^6}} \, dx\\ &=\frac {1}{6 x \sqrt {2+x^6}}-\frac {\sqrt {2+x^6}}{3 x}-\frac {1}{3} \int \frac {2^{2/3} \left (-1+\sqrt {3}\right )-2 x^4}{\sqrt {2+x^6}} \, dx-\frac {1}{3} \left (2^{2/3} \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {2+x^6}} \, dx\\ &=\frac {1}{6 x \sqrt {2+x^6}}-\frac {\sqrt {2+x^6}}{3 x}+\frac {\left (1+\sqrt {3}\right ) x \sqrt {2+x^6}}{3 \left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )}-\frac {\sqrt [3]{2} x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3^{3/4} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}}-\frac {\left (1-\sqrt {3}\right ) x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}}\\ \end {align*}

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Mathematica [C]  time = 0.00, size = 29, normalized size = 0.07 \[ -\frac {\, _2F_1\left (-\frac {1}{6},\frac {3}{2};\frac {5}{6};-\frac {x^6}{2}\right )}{2 \sqrt {2} x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(2 + x^6)^(3/2)),x]

[Out]

-1/2*Hypergeometric2F1[-1/6, 3/2, 5/6, -1/2*x^6]/(Sqrt[2]*x)

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fricas [F]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{6} + 2}}{x^{14} + 4 \, x^{8} + 4 \, x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x^6 + 2)/(x^14 + 4*x^8 + 4*x^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{6} + 2\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((x^6 + 2)^(3/2)*x^2), x)

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maple [C]  time = 0.14, size = 40, normalized size = 0.10 \[ \frac {\sqrt {2}\, x^{5} \hypergeom \left (\left [\frac {1}{2}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], -\frac {x^{6}}{2}\right )}{15}-\frac {2 x^{6}+3}{6 \sqrt {x^{6}+2}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*x^6+3)/x/(x^6+2)^(1/2)+1/15*2^(1/2)*x^5*hypergeom([1/2,5/6],[11/6],-1/2*x^6)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{6} + 2\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((x^6 + 2)^(3/2)*x^2), x)

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mupad [B]  time = 1.26, size = 31, normalized size = 0.08 \[ -\frac {{\left (\frac {2}{x^6}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {5}{3};\ \frac {8}{3};\ -\frac {2}{x^6}\right )}{10\,x\,{\left (x^6+2\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(x^6 + 2)^(3/2)),x)

[Out]

-((2/x^6 + 1)^(3/2)*hypergeom([3/2, 5/3], 8/3, -2/x^6))/(10*x*(x^6 + 2)^(3/2))

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sympy [C]  time = 1.54, size = 37, normalized size = 0.09 \[ \frac {\sqrt {2} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{2} \\ \frac {5}{6} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 x \Gamma \left (\frac {5}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*gamma(-1/6)*hyper((-1/6, 3/2), (5/6,), x**6*exp_polar(I*pi)/2)/(24*x*gamma(5/6))

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