3.6.6 \(\int \frac {x^4}{\sqrt {-1-x^3}} \, dx\) [506]

Optimal. Leaf size=264 \[ -\frac {2}{7} x^2 \sqrt {-1-x^3}+\frac {8 \sqrt {-1-x^3}}{7 \left (1-\sqrt {3}+x\right )}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{7 \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {8 \sqrt {2} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{7 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]

[Out]

-2/7*x^2*(-x^3-1)^(1/2)+8/7*(-x^3-1)^(1/2)/(1+x-3^(1/2))+8/21*(1+x)*EllipticF((1+x+3^(1/2))/(1+x-3^(1/2)),2*I-
I*3^(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)-4/7
*3^(1/4)*(1+x)*EllipticE((1+x+3^(1/2))/(1+x-3^(1/2)),2*I-I*3^(1/2))*((x^2-x+1)/(1+x-3^(1/2))^2)^(1/2)*(1/2*6^(
1/2)+1/2*2^(1/2))/(-x^3-1)^(1/2)/((-1-x)/(1+x-3^(1/2))^2)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {327, 310, 225, 1893} \begin {gather*} \frac {8 \sqrt {2} (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 )}{7 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} E\left (\text {ArcSin}\left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{7 \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {8 \sqrt {-x^3-1}}{7 \left (x-\sqrt {3}+1\right )}-\frac {2}{7} \sqrt {-x^3-1} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[-1 - x^3],x]

[Out]

(-2*x^2*Sqrt[-1 - x^3])/7 + (8*Sqrt[-1 - x^3])/(7*(1 - Sqrt[3] + x)) - (4*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 + x)*Sq
rt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/
(7*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3]) + (8*Sqrt[2]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] +
x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/(7*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 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x]
, 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]

Rule 1893

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[(1 + Sqrt[3])*(d/c)]]
, s = Denom[Simplify[(1 + Sqrt[3])*(d/c)]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x
] + Simp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/(
(1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt {-1-x^3}} \, dx &=-\frac {2}{7} x^2 \sqrt {-1-x^3}-\frac {4}{7} \int \frac {x}{\sqrt {-1-x^3}} \, dx\\ &=-\frac {2}{7} x^2 \sqrt {-1-x^3}-\frac {4}{7} \int \frac {1+\sqrt {3}+x}{\sqrt {-1-x^3}} \, dx+\frac {1}{7} \left (4 \sqrt {2 \left (2+\sqrt {3}\right )}\right ) \int \frac {1}{\sqrt {-1-x^3}} \, dx\\ &=-\frac {2}{7} x^2 \sqrt {-1-x^3}+\frac {8 \sqrt {-1-x^3}}{7 \left (1-\sqrt {3}+x\right )}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{7 \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {8 \sqrt {2} (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 )}{7 \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 = 10.02, size = 49, normalized size = 0.19 \begin {gather*} \frac {2 x^2 \left (1+x^3-\sqrt {1+x^3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-x^3\right )\right )}{7 \sqrt {-1-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[-1 - x^3],x]

[Out]

(2*x^2*(1 + x^3 - Sqrt[1 + x^3]*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3]))/(7*Sqrt[-1 - x^3])

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Maple [A]
time = 0.14, size = 175, normalized size = 0.66

method	result	size
meijerg	\(-\frac {i x^{5} \hypergeom \left (\left [\frac {1}{2}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], -x^{3}\right )}{5}\)	\(18\)
default	\(-\frac {2 x^{2} \sqrt {-x^{3}-1}}{7}+\frac {8 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-\EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{21 \sqrt {-x^{3}-1}}\)	\(175\)
elliptic	\(-\frac {2 x^{2} \sqrt {-x^{3}-1}}{7}+\frac {8 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-\EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{21 \sqrt {-x^{3}-1}}\)	\(175\)
risch	\(\frac {2 x^{2} \left (x^{3}+1\right )}{7 \sqrt {-x^{3}-1}}+\frac {8 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-\EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{21 \sqrt {-x^{3}-1}}\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/7*x^2*(-x^3-1)^(1/2)+8/21*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)*((3/2+1/2*I*3^(1/2))*EllipticE(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))-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^4/(-x^3-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4/sqrt(-x^3 - 1), x)

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Fricas [A]
time = 0.08, size = 14, normalized size = 0.05 \begin {gather*} -\frac {2}{7} \, \sqrt {-x^{3} - 1} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^3-1)^(1/2),x, algorithm="fricas")

[Out]

-2/7*sqrt(-x^3 - 1)*x^2

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Sympy [A]
time = 0.33, size = 32, normalized size = 0.12 \begin {gather*} - \frac {i x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(-x**3-1)**(1/2),x)

[Out]

-I*x**5*gamma(5/3)*hyper((1/2, 5/3), (8/3,), x**3*exp_polar(I*pi))/(3*gamma(8/3))

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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^4/(-x^3-1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(-x^3 - 1), x)

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Mupad [B]
time = 0.06, size = 245, normalized size = 0.93 \begin {gather*} -\frac {2\,x^2\,\sqrt {-x^3-1}}{7}+\frac {8\,\left (\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\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 )-\left (-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {E}\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 )\right )\,\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}}}}{7\,\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^4/(- x^3 - 1)^(1/2),x)

[Out]

(8*(((3^(1/2)*1i)/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)) - ((3^(1/2)*1i)/2 - 3/2)*ellipticE(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)))*((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))/(7*(- 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)) - (2*x^2*(- x^3 - 1)^(1/2))/7

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