\(\int \frac {x^6}{\sqrt {1-x^3}} \, dx\) [464]

Optimal result

Integrand size = 15, antiderivative size = 152 \[ \int \frac {x^6}{\sqrt {1-x^3}} \, dx=-\frac {16}{55} x \sqrt {1-x^3}-\frac {2}{11} x^4 \sqrt {1-x^3}-\frac {32 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{55 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \]

[Out]

-16/55*x*(-x^3+1)^(1/2)-2/11*x^4*(-x^3+1)^(1/2)-32/165*(1-x)*EllipticF((1-x-3^(1/2))/(1-x+3^(1/2)),I*3^(1/2)+2
*I)*(1/2*6^(1/2)+1/2*2^(1/2))*((x^2+x+1)/(1-x+3^(1/2))^2)^(1/2)*3^(3/4)/(-x^3+1)^(1/2)/((1-x)/(1-x+3^(1/2))^2)
^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {327, 224} \[ \int \frac {x^6}{\sqrt {1-x^3}} \, dx=-\frac {32 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{55 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {16}{55} \sqrt {1-x^3} x-\frac {2}{11} \sqrt {1-x^3} x^4 \]

[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 224

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

Rule 327

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]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.26 \[ \int \frac {x^6}{\sqrt {1-x^3}} \, dx=-\frac {2}{55} x \left (\sqrt {1-x^3} \left (8+5 x^3\right )-8 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},x^3\right )\right ) \]

[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 [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 4.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.10

method	result	size
meijerg	\(\frac {x^{7} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {7}{3};\frac {10}{3};x^{3}\right )}{7}\)	\(15\)
risch	\(\frac {2 x \left (5 x^{3}+8\right ) \left (x^{3}-1\right )}{55 \sqrt {-x^{3}+1}}-\frac {32 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {-1+x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{165 \sqrt {-x^{3}+1}}\)	\(132\)
default	\(-\frac {2 x^{4} \sqrt {-x^{3}+1}}{11}-\frac {16 x \sqrt {-x^{3}+1}}{55}-\frac {32 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {-1+x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{165 \sqrt {-x^{3}+1}}\)	\(134\)
elliptic	\(-\frac {2 x^{4} \sqrt {-x^{3}+1}}{11}-\frac {16 x \sqrt {-x^{3}+1}}{55}-\frac {32 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {-1+x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{165 \sqrt {-x^{3}+1}}\)	\(134\)

[In]

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

[Out]

1/7*x^7*hypergeom([1/2,7/3],[10/3],x^3)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {x^6}{\sqrt {1-x^3}} \, dx=-\frac {2}{55} \, {\left (5 \, x^{4} + 8 \, x\right )} \sqrt {-x^{3} + 1} - \frac {32}{55} i \, {\rm weierstrassPInverse}\left (0, 4, x\right ) \]

[In]

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

[Out]

-2/55*(5*x^4 + 8*x)*sqrt(-x^3 + 1) - 32/55*I*weierstrassPInverse(0, 4, x)

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.20 \[ \int \frac {x^6}{\sqrt {1-x^3}} \, dx=\frac {x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \]

[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))

Maxima [F]

\[ \int \frac {x^6}{\sqrt {1-x^3}} \, dx=\int { \frac {x^{6}}{\sqrt {-x^{3} + 1}} \,d x } \]

[In]

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

[Out]

integrate(x^6/sqrt(-x^3 + 1), x)

Giac [F]

\[ \int \frac {x^6}{\sqrt {1-x^3}} \, dx=\int { \frac {x^{6}}{\sqrt {-x^{3} + 1}} \,d x } \]

[In]

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

[Out]

integrate(x^6/sqrt(-x^3 + 1), x)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.31 \[ \int \frac {x^6}{\sqrt {1-x^3}} \, dx=-\frac {2\,x^4\,\sqrt {1-x^3}}{11}-\frac {16\,x\,\sqrt {1-x^3}}{55}-\frac {32\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3-1}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{55\,\sqrt {1-x^3}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

int(x^6/(1 - x^3)^(1/2),x)

[Out]

- (2*x^4*(1 - x^3)^(1/2))/11 - (16*x*(1 - x^3)^(1/2))/55 - (32*((3^(1/2)*1i)/2 + 3/2)*(x^3 - 1)^(1/2)*(-(x - (
3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(
-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticF(asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)
/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(55*(1 - x^3)^(1/2)*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3
^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2))