3.5.100 \(\int \frac {x^6}{\sqrt {-1-x^3}} \, dx\) [500]

Optimal. Leaf size=149 \[ \frac {16}{55} x \sqrt {-1-x^3}-\frac {2}{11} x^4 \sqrt {-1-x^3}+\frac {32 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{55 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]

[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)),2*I-I*3^(1/2
))*(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)

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Rubi [A]
time = 0.03, antiderivative size = 149, normalized size of antiderivative = 1.00, 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 = {327, 225} \begin {gather*} \frac {32 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{55 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {16}{55} \sqrt {-x^3-1} x-\frac {2}{11} \sqrt {-x^3-1} x^4 \end {gather*}

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 - S
qrt[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 225

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] && NegQ[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*} \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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 54, normalized size = 0.36 \begin {gather*} \frac {2 x \left (-8-3 x^3+5 x^6+8 \sqrt {1+x^3} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-x^3\right )\right )}{55 \sqrt {-1-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/Sqrt[-1 - x^3],x]

[Out]

(2*x*(-8 - 3*x^3 + 5*x^6 + 8*Sqrt[1 + x^3]*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3]))/(55*Sqrt[-1 - x^3])

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Maple [A]
time = 0.16, size = 134, normalized size = 0.90

method	result	size
meijerg	\(-\frac {i x^{7} \hypergeom \left (\left [\frac {1}{2}, \frac {7}{3}\right ], \left [\frac {10}{3}\right ], -x^{3}\right )}{7}\)	\(18\)
risch	\(\frac {2 x \left (5 x^{3}-8\right ) \left (x^{3}+1\right )}{55 \sqrt {-x^{3}-1}}-\frac {32 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{165 \sqrt {-x^{3}-1}}\)	\(132\)
default	\(-\frac {2 x^{4} \sqrt {-x^{3}-1}}{11}+\frac {16 x \sqrt {-x^{3}-1}}{55}-\frac {32 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{165 \sqrt {-x^{3}-1}}\)	\(134\)
elliptic	\(-\frac {2 x^{4} \sqrt {-x^{3}-1}}{11}+\frac {16 x \sqrt {-x^{3}-1}}{55}-\frac {32 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{165 \sqrt {-x^{3}-1}}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)*((x+1
)/(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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [A]
time = 0.07, size = 20, normalized size = 0.13 \begin {gather*} -\frac {2}{55} \, {\left (5 \, x^{4} - 8 \, x\right )} \sqrt {-x^{3} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/55*(5*x^4 - 8*x)*sqrt(-x^3 - 1)

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Sympy [A]
time = 0.37, size = 32, normalized size = 0.21 \begin {gather*} - \frac {i 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^{i \pi }} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.04, size = 199, normalized size = 1.34 \begin {gather*} \frac {16\,x\,\sqrt {-x^3-1}}{55}-\frac {2\,x^4\,\sqrt {-x^3-1}}{11}+\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+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\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 {-x^3-1}\,\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(- x^3 - 1)^(1/2),x)

[Out]

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

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