3.6.9 \(\int \frac {1}{x^5 \sqrt {-1-x^3}} \, dx\) [509]

Optimal. Leaf size=282 \[ \frac {\sqrt {-1-x^3}}{4 x^4}-\frac {5 \sqrt {-1-x^3}}{8 x}+\frac {5 \sqrt {-1-x^3}}{8 \left (1-\sqrt {3}+x\right )}-\frac {5 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{16 \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {5 (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{4 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]

[Out]

1/4*(-x^3-1)^(1/2)/x^4-5/8*(-x^3-1)^(1/2)/x+5/8*(-x^3-1)^(1/2)/(1+x-3^(1/2))+5/24*(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)-5/16*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 = 0.06, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {331, 310, 225, 1893} \begin {gather*} \frac {5 (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 )}{4 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {5 \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 )}{16 \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {5 \sqrt {-x^3-1}}{8 x}+\frac {5 \sqrt {-x^3-1}}{8 \left (x-\sqrt {3}+1\right )}+\frac {\sqrt {-x^3-1}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*Sqrt[-1 - x^3]),x]

[Out]

Sqrt[-1 - x^3]/(4*x^4) - (5*Sqrt[-1 - x^3])/(8*x) + (5*Sqrt[-1 - x^3])/(8*(1 - Sqrt[3] + x)) - (5*3^(1/4)*Sqrt
[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3]
+ x)], -7 + 4*Sqrt[3]])/(16*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3]) + (5*(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]])/(4*Sqrt[2]*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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 {1}{x^5 \sqrt {-1-x^3}} \, dx &=\frac {\sqrt {-1-x^3}}{4 x^4}-\frac {5}{8} \int \frac {1}{x^2 \sqrt {-1-x^3}} \, dx\\ &=\frac {\sqrt {-1-x^3}}{4 x^4}-\frac {5 \sqrt {-1-x^3}}{8 x}-\frac {5}{16} \int \frac {x}{\sqrt {-1-x^3}} \, dx\\ &=\frac {\sqrt {-1-x^3}}{4 x^4}-\frac {5 \sqrt {-1-x^3}}{8 x}-\frac {5}{16} \int \frac {1+\sqrt {3}+x}{\sqrt {-1-x^3}} \, dx+\frac {1}{8} \left (5 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )}\right ) \int \frac {1}{\sqrt {-1-x^3}} \, dx\\ &=\frac {\sqrt {-1-x^3}}{4 x^4}-\frac {5 \sqrt {-1-x^3}}{8 x}+\frac {5 \sqrt {-1-x^3}}{8 \left (1-\sqrt {3}+x\right )}-\frac {5 \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 )}{16 \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {5 (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 )}{4 \sqrt {2} \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 = 10.01, size = 42, normalized size = 0.15 \begin {gather*} -\frac {\sqrt {1+x^3} \, _2F_1\left (-\frac {4}{3},\frac {1}{2};-\frac {1}{3};-x^3\right )}{4 x^4 \sqrt {-1-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*Sqrt[-1 - x^3]),x]

[Out]

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

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Maple [A]
time = 0.14, size = 189, normalized size = 0.67

method	result	size
meijerg	\(\frac {i \hypergeom \left (\left [-\frac {4}{3}, \frac {1}{2}\right ], \left [-\frac {1}{3}\right ], -x^{3}\right )}{4 x^{4}}\)	\(18\)
risch	\(\frac {5 x^{6}+3 x^{3}-2}{8 x^{4} \sqrt {-x^{3}-1}}+\frac {5 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-\EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{24 \sqrt {-x^{3}-1}}\)	\(187\)
default	\(\frac {\sqrt {-x^{3}-1}}{4 x^{4}}-\frac {5 \sqrt {-x^{3}-1}}{8 x}+\frac {5 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-\EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{24 \sqrt {-x^{3}-1}}\)	\(189\)
elliptic	\(\frac {\sqrt {-x^{3}-1}}{4 x^{4}}-\frac {5 \sqrt {-x^{3}-1}}{8 x}+\frac {5 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-\EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{24 \sqrt {-x^{3}-1}}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(-x^3-1)^(1/2)/x^4-5/8*(-x^3-1)^(1/2)/x+5/24*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(1/x^5/(-x^3-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^3 - 1)*x^5), x)

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Fricas [A]
time = 0.08, size = 21, normalized size = 0.07 \begin {gather*} -\frac {{\left (5 \, x^{3} - 2\right )} \sqrt {-x^{3} - 1}}{8 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-x^3-1)^(1/2),x, algorithm="fricas")

[Out]

-1/8*(5*x^3 - 2)*sqrt(-x^3 - 1)/x^4

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Sympy [A]
time = 0.41, size = 39, normalized size = 0.14 \begin {gather*} - \frac {i \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(-x**3-1)**(1/2),x)

[Out]

-I*gamma(-4/3)*hyper((-4/3, 1/2), (-1/3,), x**3*exp_polar(I*pi))/(3*x**4*gamma(-1/3))

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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(1/x^5/(-x^3-1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^3 - 1)*x^5), x)

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Mupad [B]
time = 0.05, size = 259, normalized size = 0.92 \begin {gather*} \frac {\sqrt {-x^3-1}}{4\,x^4}-\frac {5\,\sqrt {-x^3-1}}{8\,x}+\frac {5\,\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}}}}{8\,\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(1/(x^5*(- x^3 - 1)^(1/2)),x)

[Out]

(- x^3 - 1)^(1/2)/(4*x^4) - (5*(- x^3 - 1)^(1/2))/(8*x) + (5*(((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)*ellipt
icE(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)*1
i)/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))/(8*(- 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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