3.1404 \(\int \frac{1}{x^3 \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=378 \[ \frac{\left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right ),-7-4 \sqrt{3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}+\frac{\sqrt{x^6+2}}{4 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{\sqrt{x^6+2}}{4 x^2}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{4\ 2^{5/6} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

[Out]

-Sqrt[2 + x^6]/(4*x^2) + Sqrt[2 + x^6]/(4*(2^(1/3)*(1 + Sqrt[3]) + x^2)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*(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[ArcSin[(2^(1/3)*(1 - Sqr
t[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(4*2^(5/6)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sq
rt[3]) + x^2)^2]*Sqrt[2 + x^6]) + ((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[ArcSin[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(2*2^
(1/3)*3^(1/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6])

________________________________________________________________________________________

Rubi [A]  time = 0.192884, antiderivative size = 378, normalized size of antiderivative = 1., 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 = {275, 325, 303, 218, 1877} \[ \frac{\sqrt{x^6+2}}{4 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{\sqrt{x^6+2}}{4 x^2}+\frac{\left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{4\ 2^{5/6} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(4*x^2) + Sqrt[2 + x^6]/(4*(2^(1/3)*(1 + Sqrt[3]) + x^2)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*(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[ArcSin[(2^(1/3)*(1 - Sqr
t[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(4*2^(5/6)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sq
rt[3]) + x^2)^2]*Sqrt[2 + x^6]) + ((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[ArcSin[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(2*2^
(1/3)*3^(1/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6])

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 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 303

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

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \sqrt{2+x^6}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{2+x^3}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{2+x^6}}{4 x^2}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{x}{\sqrt{2+x^3}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{2+x^6}}{4 x^2}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{\sqrt [3]{2} \left (1-\sqrt{3}\right )+x}{\sqrt{2+x^3}} \, dx,x,x^2\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2+x^3}} \, dx,x,x^2\right )}{4 \sqrt [6]{2} \sqrt{2+\sqrt{3}}}\\ &=-\frac{\sqrt{2+x^6}}{4 x^2}+\frac{\sqrt{2+x^6}}{4 \left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \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 (\sin ^{-1}\left (\frac{\sqrt [3]{2} \left (1-\sqrt{3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2}\right )|-7-4 \sqrt{3}\right )}{4\ 2^{5/6} \sqrt{\frac{\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )^2}} \sqrt{2+x^6}}+\frac{\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 (\sin ^{-1}\left (\frac{\sqrt [3]{2} \left (1-\sqrt{3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )^2}} \sqrt{2+x^6}}\\ \end{align*}

Mathematica [C]  time = 0.0047197, size = 29, normalized size = 0.08 \[ -\frac{\, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};-\frac{x^6}{2}\right )}{2 \sqrt{2} x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[2 + x^6]),x]

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.027, size = 33, normalized size = 0.1 \begin{align*} -{\frac{1}{4\,{x}^{2}}\sqrt{{x}^{6}+2}}+{\frac{{x}^{4}\sqrt{2}}{32}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{2}{3}};\,{\frac{5}{3}};\,-{\frac{{x}^{6}}{2}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(x^6+2)^(1/2)/x^2+1/32*2^(1/2)*x^4*hypergeom([1/2,2/3],[5/3],-1/2*x^6)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{6} + 2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x^6 + 2)*x^3), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{6} + 2}}{x^{9} + 2 \, x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x^6 + 2)/(x^9 + 2*x^3), x)

________________________________________________________________________________________

Sympy [A]  time = 0.640571, size = 39, normalized size = 0.1 \begin{align*} \frac{\sqrt{2} \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{2} \\ \frac{2}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{12 x^{2} \Gamma \left (\frac{2}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*gamma(-1/3)*hyper((-1/3, 1/2), (2/3,), x**6*exp_polar(I*pi)/2)/(12*x**2*gamma(2/3))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{6} + 2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x^6 + 2)*x^3), x)