3.1424 \(\int \frac{x^{12}}{(2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=195 \[ -\frac{7 \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x \text{EllipticF}\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}-\frac{x^7}{3 \sqrt{x^6+2}}+\frac{7}{12} \sqrt{x^6+2} x \]

[Out]

-x^7/(3*Sqrt[2 + x^6]) + (7*x*Sqrt[2 + x^6])/12 - (7*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])/(12*2^(1/3)*3^(1/4)*Sqrt[(x^2*(2^(1/3) + x^2))/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*Sqrt[2 + x^6])

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Rubi [A]  time = 0.0454726, antiderivative size = 195, 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 = {288, 321, 225} \[ -\frac{x^7}{3 \sqrt{x^6+2}}+\frac{7}{12} \sqrt{x^6+2} x-\frac{7 \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Int[x^12/(2 + x^6)^(3/2),x]

[Out]

-x^7/(3*Sqrt[2 + x^6]) + (7*x*Sqrt[2 + x^6])/12 - (7*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])/(12*2^(1/3)*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 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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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]

Rubi steps

\begin{align*} \int \frac{x^{12}}{\left (2+x^6\right )^{3/2}} \, dx &=-\frac{x^7}{3 \sqrt{2+x^6}}+\frac{7}{3} \int \frac{x^6}{\sqrt{2+x^6}} \, dx\\ &=-\frac{x^7}{3 \sqrt{2+x^6}}+\frac{7}{12} x \sqrt{2+x^6}-\frac{7}{6} \int \frac{1}{\sqrt{2+x^6}} \, dx\\ &=-\frac{x^7}{3 \sqrt{2+x^6}}+\frac{7}{12} x \sqrt{2+x^6}-\frac{7 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 )}{12 \sqrt [3]{2} \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*}

Mathematica [C]  time = 0.011715, size = 54, normalized size = 0.28 \[ \frac{x \left (-7 \sqrt{2} \sqrt{x^6+2} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{x^6}{2}\right )+3 x^6+14\right )}{12 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^12/(2 + x^6)^(3/2),x]

[Out]

(x*(14 + 3*x^6 - 7*Sqrt[2]*Sqrt[2 + x^6]*Hypergeometric2F1[1/6, 1/2, 7/6, -x^6/2]))/(12*Sqrt[2 + x^6])

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Maple [C]  time = 0.024, size = 36, normalized size = 0.2 \begin{align*}{\frac{x \left ( 3\,{x}^{6}+14 \right ) }{12}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{7\,x\sqrt{2}}{12}{\mbox{$_2$F$_1$}({\frac{1}{6}},{\frac{1}{2}};\,{\frac{7}{6}};\,-{\frac{{x}^{6}}{2}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^12/(x^6+2)^(3/2),x)

[Out]

1/12*x*(3*x^6+14)/(x^6+2)^(1/2)-7/12*2^(1/2)*x*hypergeom([1/6,1/2],[7/6],-1/2*x^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{12}}{{\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^12/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^12/(x^6 + 2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{6} + 2} x^{12}}{x^{12} + 4 \, x^{6} + 4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(x^6 + 2)*x^12/(x^12 + 4*x^6 + 4), x)

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Sympy [C]  time = 1.36191, size = 36, normalized size = 0.18 \begin{align*} \frac{\sqrt{2} x^{13} \Gamma \left (\frac{13}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{13}{6} \\ \frac{19}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{19}{6}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*x**13*gamma(13/6)*hyper((3/2, 13/6), (19/6,), x**6*exp_polar(I*pi)/2)/(24*gamma(19/6))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{12}}{{\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^12/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^12/(x^6 + 2)^(3/2), x)