3.464 \(\int \frac{x^6}{\sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{32 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{2}{11} \sqrt{1-x^3} x^4-\frac{16}{55} \sqrt{1-x^3} x \]

[Out]

(-16*x*Sqrt[1 - x^3])/55 - (2*x^4*Sqrt[1 - x^3])/11 - (32*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sq
rt[3] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(55*3^(1/4)*Sqrt[(1 - x)
/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])

________________________________________________________________________________________

Rubi [A]  time = 0.0381576, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {321, 218} \[ -\frac{2}{11} \sqrt{1-x^3} x^4-\frac{16}{55} \sqrt{1-x^3} x-\frac{32 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

[In]

Int[x^6/Sqrt[1 - x^3],x]

[Out]

(-16*x*Sqrt[1 - x^3])/55 - (2*x^4*Sqrt[1 - x^3])/11 - (32*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sq
rt[3] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(55*3^(1/4)*Sqrt[(1 - x)
/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])

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 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]

Rubi steps

\begin{align*} \int \frac{x^6}{\sqrt{1-x^3}} \, dx &=-\frac{2}{11} x^4 \sqrt{1-x^3}+\frac{8}{11} \int \frac{x^3}{\sqrt{1-x^3}} \, dx\\ &=-\frac{16}{55} x \sqrt{1-x^3}-\frac{2}{11} x^4 \sqrt{1-x^3}+\frac{16}{55} \int \frac{1}{\sqrt{1-x^3}} \, dx\\ &=-\frac{16}{55} x \sqrt{1-x^3}-\frac{2}{11} x^4 \sqrt{1-x^3}-\frac{32 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}-x}{1+\sqrt{3}-x}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}} \sqrt{1-x^3}}\\ \end{align*}

Mathematica [C]  time = 0.0142566, size = 40, normalized size = 0.26 \[ -\frac{2}{55} x \left (\sqrt{1-x^3} \left (5 x^3+8\right )-8 \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/Sqrt[1 - x^3],x]

[Out]

(-2*x*(Sqrt[1 - x^3]*(8 + 5*x^3) - 8*Hypergeometric2F1[1/3, 1/2, 4/3, x^3]))/55

________________________________________________________________________________________

Maple [A]  time = 0.022, size = 134, normalized size = 0.9 \begin{align*} -{\frac{2\,{x}^{4}}{11}\sqrt{-{x}^{3}+1}}-{\frac{16\,x}{55}\sqrt{-{x}^{3}+1}}-{{\frac{32\,i}{165}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*x^4*(-x^3+1)^(1/2)-16/55*x*(-x^3+1)^(1/2)-32/165*I*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((-1+
x)/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)*EllipticF(1/3*3^(1/2)*(
I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^6/sqrt(-x^3 + 1), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{3} + 1} x^{6}}{x^{3} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-x^3 + 1)*x^6/(x^3 - 1), x)

________________________________________________________________________________________

Sympy [A]  time = 0.897247, size = 31, normalized size = 0.2 \begin{align*} \frac{x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**7*gamma(7/3)*hyper((1/2, 7/3), (10/3,), x**3*exp_polar(2*I*pi))/(3*gamma(10/3))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^6/sqrt(-x^3 + 1), x)