3.10.3 \(\int \frac {\sqrt {-1+x^3} (1-x^3+x^6)}{x^{10} (2+x^3)} \, dx\)

Optimal. Leaf size=68 \[ \frac {13}{24} \tan ^{-1}\left (\sqrt {x^3-1}\right )-\frac {7 \tan ^{-1}\left (\frac {\sqrt {x^3-1}}{\sqrt {3}}\right )}{8 \sqrt {3}}+\frac {\sqrt {x^3-1} \left (-12 x^6+5 x^3-2\right )}{36 x^9} \]

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Rubi [A]  time = 0.51, antiderivative size = 88, normalized size of antiderivative = 1.29, number of steps used = 25, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6725, 266, 47, 51, 63, 203, 50, 444} \begin {gather*} -\frac {\sqrt {x^3-1}}{3 x^3}+\frac {13}{24} \tan ^{-1}\left (\sqrt {x^3-1}\right )-\frac {7 \tan ^{-1}\left (\frac {\sqrt {x^3-1}}{\sqrt {3}}\right )}{8 \sqrt {3}}-\frac {\sqrt {x^3-1}}{18 x^9}+\frac {5 \sqrt {x^3-1}}{36 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-1 + x^3]*(1 - x^3 + x^6))/(x^10*(2 + x^3)),x]

[Out]

-1/18*Sqrt[-1 + x^3]/x^9 + (5*Sqrt[-1 + x^3])/(36*x^6) - Sqrt[-1 + x^3]/(3*x^3) + (13*ArcTan[Sqrt[-1 + x^3]])/
24 - (7*ArcTan[Sqrt[-1 + x^3]/Sqrt[3]])/(8*Sqrt[3])

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x^3} \left (1-x^3+x^6\right )}{x^{10} \left (2+x^3\right )} \, dx &=\int \left (\frac {\sqrt {-1+x^3}}{2 x^{10}}-\frac {3 \sqrt {-1+x^3}}{4 x^7}+\frac {7 \sqrt {-1+x^3}}{8 x^4}-\frac {7 \sqrt {-1+x^3}}{16 x}+\frac {7 x^2 \sqrt {-1+x^3}}{16 \left (2+x^3\right )}\right ) \, dx\\ &=-\left (\frac {7}{16} \int \frac {\sqrt {-1+x^3}}{x} \, dx\right )+\frac {7}{16} \int \frac {x^2 \sqrt {-1+x^3}}{2+x^3} \, dx+\frac {1}{2} \int \frac {\sqrt {-1+x^3}}{x^{10}} \, dx-\frac {3}{4} \int \frac {\sqrt {-1+x^3}}{x^7} \, dx+\frac {7}{8} \int \frac {\sqrt {-1+x^3}}{x^4} \, dx\\ &=-\left (\frac {7}{48} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^3\right )\right )+\frac {7}{48} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{2+x} \, dx,x,x^3\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^4} \, dx,x,x^3\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^3} \, dx,x,x^3\right )+\frac {7}{24} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {-1+x^3}}{18 x^9}+\frac {\sqrt {-1+x^3}}{8 x^6}-\frac {7 \sqrt {-1+x^3}}{24 x^3}+\frac {1}{36} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^3} \, dx,x,x^3\right )-\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^3\right )+2 \left (\frac {7}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right )\right )-\frac {7}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} (2+x)} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {-1+x^3}}{18 x^9}+\frac {5 \sqrt {-1+x^3}}{36 x^6}-\frac {17 \sqrt {-1+x^3}}{48 x^3}+\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^3\right )-\frac {1}{32} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right )+2 \left (\frac {7}{24} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right )\right )-\frac {7}{8} \operatorname {Subst}\left (\int \frac {1}{3+x^2} \, dx,x,\sqrt {-1+x^3}\right )\\ &=-\frac {\sqrt {-1+x^3}}{18 x^9}+\frac {5 \sqrt {-1+x^3}}{36 x^6}-\frac {\sqrt {-1+x^3}}{3 x^3}+\frac {7}{12} \tan ^{-1}\left (\sqrt {-1+x^3}\right )-\frac {7 \tan ^{-1}\left (\frac {\sqrt {-1+x^3}}{\sqrt {3}}\right )}{8 \sqrt {3}}+\frac {1}{96} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right )-\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right )\\ &=-\frac {\sqrt {-1+x^3}}{18 x^9}+\frac {5 \sqrt {-1+x^3}}{36 x^6}-\frac {\sqrt {-1+x^3}}{3 x^3}+\frac {25}{48} \tan ^{-1}\left (\sqrt {-1+x^3}\right )-\frac {7 \tan ^{-1}\left (\frac {\sqrt {-1+x^3}}{\sqrt {3}}\right )}{8 \sqrt {3}}+\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right )\\ &=-\frac {\sqrt {-1+x^3}}{18 x^9}+\frac {5 \sqrt {-1+x^3}}{36 x^6}-\frac {\sqrt {-1+x^3}}{3 x^3}+\frac {13}{24} \tan ^{-1}\left (\sqrt {-1+x^3}\right )-\frac {7 \tan ^{-1}\left (\frac {\sqrt {-1+x^3}}{\sqrt {3}}\right )}{8 \sqrt {3}}\\ \end {align*}

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Mathematica [C]  time = 0.18, size = 146, normalized size = 2.15 \begin {gather*} \frac {-12 \left (x^3-1\right )^2 \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};1-x^3\right )+8 \left (x^3-1\right )^2 \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};1-x^3\right )+21 \left (\frac {1}{x^3}+\sqrt {x^3-1} \tan ^{-1}\left (\sqrt {x^3-1}\right )-\sqrt {3} \sqrt {x^3-1} \tan ^{-1}\left (\frac {\sqrt {x^3-1}}{\sqrt {3}}\right )-\sqrt {1-x^3} \tanh ^{-1}\left (\sqrt {1-x^3}\right )-1\right )}{72 \sqrt {x^3-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[-1 + x^3]*(1 - x^3 + x^6))/(x^10*(2 + x^3)),x]

[Out]

(21*(-1 + x^(-3) + Sqrt[-1 + x^3]*ArcTan[Sqrt[-1 + x^3]] - Sqrt[3]*Sqrt[-1 + x^3]*ArcTan[Sqrt[-1 + x^3]/Sqrt[3
]] - Sqrt[1 - x^3]*ArcTanh[Sqrt[1 - x^3]]) - 12*(-1 + x^3)^2*Hypergeometric2F1[3/2, 3, 5/2, 1 - x^3] + 8*(-1 +
x^3)^2*Hypergeometric2F1[3/2, 4, 5/2, 1 - x^3])/(72*Sqrt[-1 + x^3])

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IntegrateAlgebraic [A]  time = 0.09, size = 68, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^3} \left (-2+5 x^3-12 x^6\right )}{36 x^9}+\frac {13}{24} \tan ^{-1}\left (\sqrt {-1+x^3}\right )-\frac {7 \tan ^{-1}\left (\frac {\sqrt {-1+x^3}}{\sqrt {3}}\right )}{8 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-1 + x^3]*(1 - x^3 + x^6))/(x^10*(2 + x^3)),x]

[Out]

(Sqrt[-1 + x^3]*(-2 + 5*x^3 - 12*x^6))/(36*x^9) + (13*ArcTan[Sqrt[-1 + x^3]])/24 - (7*ArcTan[Sqrt[-1 + x^3]/Sq
rt[3]])/(8*Sqrt[3])

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fricas [A]  time = 0.48, size = 61, normalized size = 0.90 \begin {gather*} -\frac {21 \, \sqrt {3} x^{9} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{3} - 1}\right ) - 39 \, x^{9} \arctan \left (\sqrt {x^{3} - 1}\right ) + 2 \, {\left (12 \, x^{6} - 5 \, x^{3} + 2\right )} \sqrt {x^{3} - 1}}{72 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(1/2)*(x^6-x^3+1)/x^10/(x^3+2),x, algorithm="fricas")

[Out]

-1/72*(21*sqrt(3)*x^9*arctan(1/3*sqrt(3)*sqrt(x^3 - 1)) - 39*x^9*arctan(sqrt(x^3 - 1)) + 2*(12*x^6 - 5*x^3 + 2
)*sqrt(x^3 - 1))/x^9

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giac [A]  time = 0.19, size = 62, normalized size = 0.91 \begin {gather*} -\frac {7}{24} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{3} - 1}\right ) - \frac {12 \, {\left (x^{3} - 1\right )}^{\frac {5}{2}} + 19 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 9 \, \sqrt {x^{3} - 1}}{36 \, x^{9}} + \frac {13}{24} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(1/2)*(x^6-x^3+1)/x^10/(x^3+2),x, algorithm="giac")

[Out]

-7/24*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(x^3 - 1)) - 1/36*(12*(x^3 - 1)^(5/2) + 19*(x^3 - 1)^(3/2) + 9*sqrt(x^3 -
1))/x^9 + 13/24*arctan(sqrt(x^3 - 1))

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maple [C]  time = 0.99, size = 113, normalized size = 1.66

method	result	size
trager	\(-\frac {\left (12 x^{6}-5 x^{3}+2\right ) \sqrt {x^{3}-1}}{36 x^{9}}-\frac {13 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x^{3}-1}}{x^{3}}\right )}{48}-\frac {7 \RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+3\right )-6 \sqrt {x^{3}-1}}{x^{3}+2}\right )}{48}\)	\(113\)
risch	\(-\frac {12 x^{9}-17 x^{6}+7 x^{3}-2}{36 x^{9} \sqrt {x^{3}-1}}+\frac {13 \arctan \left (\sqrt {x^{3}-1}\right )}{24}-\frac {7 \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{6}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{48}\)	\(211\)
default	\(-\frac {\sqrt {x^{3}-1}}{18 x^{9}}+\frac {5 \sqrt {x^{3}-1}}{36 x^{6}}-\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {13 \arctan \left (\sqrt {x^{3}-1}\right )}{24}-\frac {7 \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{6}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{48}\)	\(218\)
elliptic	\(-\frac {\sqrt {x^{3}-1}}{18 x^{9}}+\frac {5 \sqrt {x^{3}-1}}{36 x^{6}}-\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {13 \arctan \left (\sqrt {x^{3}-1}\right )}{24}+\frac {7 \sqrt {2}\, \left (3+i \sqrt {3}\right ) \sqrt {-\frac {-1+x}{3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x -1}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\EllipticPi \left (\sqrt {-\frac {2 \left (-1+x \right )}{3+i \sqrt {3}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{6}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {-\frac {3+i \sqrt {3}}{-3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}\right )}{48 \sqrt {x^{3}-1}}+\frac {7 \sqrt {2}\, \left (3+i \sqrt {3}\right ) \sqrt {-\frac {-1+x}{3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x -1}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\EllipticPi \left (\sqrt {-\frac {2 \left (-1+x \right )}{3+i \sqrt {3}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{6}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {-\frac {3+i \sqrt {3}}{-3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha \right )}{48 \sqrt {x^{3}-1}}+\frac {7 \sqrt {2}\, \left (3+i \sqrt {3}\right ) \sqrt {-\frac {-1+x}{3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x -1}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\EllipticPi \left (\sqrt {-\frac {2 \left (-1+x \right )}{3+i \sqrt {3}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{6}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {-\frac {3+i \sqrt {3}}{-3+i \sqrt {3}}}\right )\right )}{48 \sqrt {x^{3}-1}}\)	\(552\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(1/2)*(x^6-x^3+1)/x^10/(x^3+2),x,method=_RETURNVERBOSE)

[Out]

-1/36*(12*x^6-5*x^3+2)/x^9*(x^3-1)^(1/2)-13/48*RootOf(_Z^2+1)*ln(-(RootOf(_Z^2+1)*x^3-2*RootOf(_Z^2+1)+2*(x^3-
1)^(1/2))/x^3)-7/48*RootOf(_Z^2+3)*ln((RootOf(_Z^2+3)*x^3-4*RootOf(_Z^2+3)-6*(x^3-1)^(1/2))/(x^3+2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - x^{3} + 1\right )} \sqrt {x^{3} - 1}}{{\left (x^{3} + 2\right )} x^{10}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(1/2)*(x^6-x^3+1)/x^10/(x^3+2),x, algorithm="maxima")

[Out]

integrate((x^6 - x^3 + 1)*sqrt(x^3 - 1)/((x^3 + 2)*x^10), x)

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mupad [B]  time = 2.51, size = 107, normalized size = 1.57 \begin {gather*} \frac {5\,\sqrt {x^3-1}}{36\,x^6}-\frac {\sqrt {x^3-1}}{3\,x^3}-\frac {\sqrt {x^3-1}}{18\,x^9}+\frac {\ln \left (\frac {\left (\sqrt {x^3-1}-\mathrm {i}\right )\,{\left (\sqrt {x^3-1}+1{}\mathrm {i}\right )}^3}{x^6}\right )\,13{}\mathrm {i}}{48}+\frac {\sqrt {3}\,\ln \left (\frac {6\,\sqrt {x^3-1}-\sqrt {3}\,4{}\mathrm {i}+\sqrt {3}\,x^3\,1{}\mathrm {i}}{x^3+2}\right )\,7{}\mathrm {i}}{48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(1/2)*(x^6 - x^3 + 1))/(x^10*(x^3 + 2)),x)

[Out]

(log((((x^3 - 1)^(1/2) - 1i)*((x^3 - 1)^(1/2) + 1i)^3)/x^6)*13i)/48 + (3^(1/2)*log((3^(1/2)*x^3*1i - 3^(1/2)*4
i + 6*(x^3 - 1)^(1/2))/(x^3 + 2))*7i)/48 - (x^3 - 1)^(1/2)/(3*x^3) + (5*(x^3 - 1)^(1/2))/(36*x^6) - (x^3 - 1)^
(1/2)/(18*x^9)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)**(1/2)*(x**6-x**3+1)/x**10/(x**3+2),x)

[Out]

Timed out

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