3.1433 \(\int \frac{x^4}{(2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=391 \[ \frac{\left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x \text{EllipticF}\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{6\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}+\frac{x^5}{6 \sqrt{x^6+2}}-\frac{\left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{6 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}+\frac{\left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} 3^{3/4} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.110887, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {290, 308, 225, 1881} \[ \frac{x^5}{6 \sqrt{x^6+2}}-\frac{\left (1+\sqrt{3}\right ) \sqrt{x^6+2} x}{6 \left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )}+\frac{\left (1-\sqrt{3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{6\ 2^{2/3} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}+\frac{\left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} 3^{3/4} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(2 + x^6)^(3/2),x]

[Out]

x^5/(6*Sqrt[2 + x^6]) - ((1 + Sqrt[3])*x*Sqrt[2 + x^6])/(6*(2^(1/3) + (1 + Sqrt[3])*x^2)) + (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[ArcCos[(2^(1/3) + (1 - Sqrt[3])
*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(2^(2/3)*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])/(6*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 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 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 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{x^4}{\left (2+x^6\right )^{3/2}} \, dx &=\frac{x^5}{6 \sqrt{2+x^6}}-\frac{1}{3} \int \frac{x^4}{\sqrt{2+x^6}} \, dx\\ &=\frac{x^5}{6 \sqrt{2+x^6}}+\frac{1}{6} \int \frac{2^{2/3} \left (-1+\sqrt{3}\right )-2 x^4}{\sqrt{2+x^6}} \, dx+\frac{\left (1-\sqrt{3}\right ) \int \frac{1}{\sqrt{2+x^6}} \, dx}{3 \sqrt [3]{2}}\\ &=\frac{x^5}{6 \sqrt{2+x^6}}-\frac{\left (1+\sqrt{3}\right ) x \sqrt{2+x^6}}{6 \left (\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2\right )}+\frac{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 )}{2^{2/3} 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 )}{6\ 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*}

Mathematica [C]  time = 0.0046646, size = 29, normalized size = 0.07 \[ \frac{x^5 \, _2F_1\left (\frac{5}{6},\frac{3}{2};\frac{11}{6};-\frac{x^6}{2}\right )}{10 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(2 + x^6)^(3/2),x]

[Out]

(x^5*Hypergeometric2F1[5/6, 3/2, 11/6, -x^6/2])/(10*Sqrt[2])

________________________________________________________________________________________

Maple [C]  time = 0.024, size = 33, normalized size = 0.1 \begin{align*}{\frac{{x}^{5}}{6}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{{x}^{5}\sqrt{2}}{30}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{5}{6}};\,{\frac{11}{6}};\,-{\frac{{x}^{6}}{2}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^6+2)^(3/2),x)

[Out]

1/6*x^5/(x^6+2)^(1/2)-1/30*2^(1/2)*x^5*hypergeom([1/2,5/6],[11/6],-1/2*x^6)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\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^4/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^4/(x^6 + 2)^(3/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 0.625788, size = 36, normalized size = 0.09 \begin{align*} \frac{\sqrt{2} x^{5} \Gamma \left (\frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{6}, \frac{3}{2} \\ \frac{11}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{11}{6}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*x**5*gamma(5/6)*hyper((5/6, 3/2), (11/6,), x**6*exp_polar(I*pi)/2)/(24*gamma(11/6))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\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^4/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^4/(x^6 + 2)^(3/2), x)