3.26.43 \(\int \frac {1}{x^6 (-1+x^3) \sqrt [3]{x^2+x^3}} \, dx\)

Optimal. Leaf size=214 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {\log \left (2^{2/3} \sqrt [3]{x^3+x^2}-2 x\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3+x^2} x+\sqrt [3]{2} \left (x^3+x^2\right )^{2/3}\right )}{6 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+x^2}+x}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {3 \left (x^3+x^2\right )^{2/3} \left (4491 x^5-2994 x^4+2495 x^3+3600 x^2-3300 x+3080\right )}{52360 x^7} \]

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Rubi [C]  time = 1.28, antiderivative size = 983, normalized size of antiderivative = 4.59, number of steps used = 27, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2056, 6725, 129, 155, 12, 91} \begin {gather*} -\frac {\left (8689+731 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}-\frac {\left (8689-731 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}+\frac {2099 (x+1)}{13090 x \sqrt [3]{x^3+x^2}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {173 (x+1)}{5236 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (163+221 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}+\frac {\left (163-221 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}+\frac {107 (x+1)}{1309 x^3 \sqrt [3]{x^3+x^2}}-\frac {\left (15-17 (-1)^{2/3}\right ) (x+1)}{238 x^4 \sqrt [3]{x^3+x^2}}-\frac {\left (15+17 \sqrt [3]{-1}\right ) (x+1)}{238 x^4 \sqrt [3]{x^3+x^2}}+\frac {x+1}{119 x^4 \sqrt [3]{x^3+x^2}}+\frac {3 (x+1)}{17 x^5 \sqrt [3]{x^3+x^2}}-\frac {\left (113+23987 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}-\frac {\left (113-23987 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}+\frac {6793 (x+1)}{26180 \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) \sqrt [3]{x+1}}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \log (1-x) \sqrt [3]{x+1}}{6 \sqrt [3]{2} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \log \left (\sqrt [3]{-1} x+1\right ) \sqrt [3]{x+1}}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \log \left (1-(-1)^{2/3} x\right ) \sqrt [3]{x+1}}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{2}}-\sqrt [3]{x}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{2} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6793*(1 + x))/(26180*(x^2 + x^3)^(1/3)) - ((113 - (23987*I)*Sqrt[3])*(1 + x))/(104720*(x^2 + x^3)^(1/3)) - ((
113 + (23987*I)*Sqrt[3])*(1 + x))/(104720*(x^2 + x^3)^(1/3)) + (3*(1 + x))/(17*x^5*(x^2 + x^3)^(1/3)) + (1 + x
)/(119*x^4*(x^2 + x^3)^(1/3)) - ((15 + 17*(-1)^(1/3))*(1 + x))/(238*x^4*(x^2 + x^3)^(1/3)) - ((15 - 17*(-1)^(2
/3))*(1 + x))/(238*x^4*(x^2 + x^3)^(1/3)) + (107*(1 + x))/(1309*x^3*(x^2 + x^3)^(1/3)) + ((163 - (221*I)*Sqrt[
3])*(1 + x))/(2618*x^3*(x^2 + x^3)^(1/3)) + ((163 + (221*I)*Sqrt[3])*(1 + x))/(2618*x^3*(x^2 + x^3)^(1/3)) + (
173*(1 + x))/(5236*x^2*(x^2 + x^3)^(1/3)) + ((1151 - (1989*I)*Sqrt[3])*(1 + x))/(20944*x^2*(x^2 + x^3)^(1/3))
+ ((1151 + (1989*I)*Sqrt[3])*(1 + x))/(20944*x^2*(x^2 + x^3)^(1/3)) + (2099*(1 + x))/(13090*x*(x^2 + x^3)^(1/3
)) - ((8689 - (731*I)*Sqrt[3])*(1 + x))/(52360*x*(x^2 + x^3)^(1/3)) - ((8689 + (731*I)*Sqrt[3])*(1 + x))/(5236
0*x*(x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*ArcTan[1/Sqrt[3] + (2^(2/3)*(1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])
/(2^(1/3)*Sqrt[3]*(x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 + x)^(1/3))/(Sqrt[3]*(1
- (-1)^(1/3))^(1/3)*x^(1/3))])/(Sqrt[3]*(1 - (-1)^(1/3))^(1/3)*(x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*Ar
cTan[1/Sqrt[3] + (2*(1 + x)^(1/3))/(Sqrt[3]*(1 + (-1)^(2/3))^(1/3)*x^(1/3))])/(Sqrt[3]*(1 + (-1)^(2/3))^(1/3)*
(x^2 + x^3)^(1/3)) - (x^(2/3)*(1 + x)^(1/3)*Log[1 - x])/(6*2^(1/3)*(x^2 + x^3)^(1/3)) - (x^(2/3)*(1 + x)^(1/3)
*Log[1 + (-1)^(1/3)*x])/(6*(1 - (-1)^(1/3))^(1/3)*(x^2 + x^3)^(1/3)) - (x^(2/3)*(1 + x)^(1/3)*Log[1 - (-1)^(2/
3)*x])/(6*(1 + (-1)^(2/3))^(1/3)*(x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*Log[-x^(1/3) + (1 + x)^(1/3)/2^(1
/3)])/(2*2^(1/3)*(x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*Log[-x^(1/3) + (1 + x)^(1/3)/(1 - (-1)^(1/3))^(1/
3)])/(2*(1 - (-1)^(1/3))^(1/3)*(x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*Log[-x^(1/3) + (1 + x)^(1/3)/(1 + (
-1)^(2/3))^(1/3)])/(2*(1 + (-1)^(2/3))^(1/3)*(x^2 + x^3)^(1/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 129

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

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 {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x} \left (-1+x^3\right )} \, dx}{\sqrt [3]{x^2+x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \left (-\frac {1}{3 (1-x) x^{20/3} \sqrt [3]{1+x}}-\frac {1}{3 x^{20/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )}-\frac {1}{3 x^{20/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [3]{x^2+x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{(1-x) x^{20/3} \sqrt [3]{1+x}} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{x^2+x^3}}\\ &=\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {2}{3}-5 x}{(1-x) x^{17/3} \sqrt [3]{1+x}} \, dx}{17 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{3} \left (15+17 \sqrt [3]{-1}\right )+5 \sqrt [3]{-1} x}{x^{17/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{17 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{3} \left (15-17 (-1)^{2/3}\right )-5 (-1)^{2/3} x}{x^{17/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{17 \sqrt [3]{x^2+x^3}}\\ &=\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {214}{9}+\frac {8 x}{3}}{(1-x) x^{14/3} \sqrt [3]{1+x}} \, dx}{238 \sqrt [3]{x^2+x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{9} \left (163+221 i \sqrt {3}\right )-\frac {4}{3} \left (1-16 i \sqrt {3}\right ) x}{x^{14/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{238 \sqrt [3]{x^2+x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{9} \left (163-221 i \sqrt {3}\right )-\frac {4}{3} \left (1+16 i \sqrt {3}\right ) x}{x^{14/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{238 \sqrt [3]{x^2+x^3}}\\ &=\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {692}{27}-\frac {214 x}{3}}{(1-x) x^{11/3} \sqrt [3]{1+x}} \, dx}{2618 \sqrt [3]{x^2+x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{27} \left (-1151+1989 i \sqrt {3}\right )-\frac {2}{3} \left (125-96 i \sqrt {3}\right ) x}{x^{11/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{2618 \sqrt [3]{x^2+x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{27} \left (-1151-1989 i \sqrt {3}\right )-\frac {2}{3} \left (125+96 i \sqrt {3}\right ) x}{x^{11/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{2618 \sqrt [3]{x^2+x^3}}\\ &=\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {16792}{81}+\frac {1384 x}{27}}{(1-x) x^{8/3} \sqrt [3]{1+x}} \, dx}{20944 \sqrt [3]{x^2+x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {2}{81} \left (8689-731 i \sqrt {3}\right )-\frac {2}{27} \left (3559-419 i \sqrt {3}\right ) x}{x^{8/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{20944 \sqrt [3]{x^2+x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {2}{81} \left (8689+731 i \sqrt {3}\right )-\frac {2}{27} \left (3559+419 i \sqrt {3}\right ) x}{x^{8/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{20944 \sqrt [3]{x^2+x^3}}\\ &=\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}+\frac {\left (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {54344}{243}-\frac {16792 x}{81}}{(1-x) x^{5/3} \sqrt [3]{1+x}} \, dx}{104720 \sqrt [3]{x^2+x^3}}+\frac {\left (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{243} \left (113+23987 i \sqrt {3}\right )-\frac {2}{81} \left (5441-3979 i \sqrt {3}\right ) x}{x^{5/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{104720 \sqrt [3]{x^2+x^3}}+\frac {\left (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{243} \left (113-23987 i \sqrt {3}\right )-\frac {2}{81} \left (5441+3979 i \sqrt {3}\right ) x}{x^{5/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{104720 \sqrt [3]{x^2+x^3}}\\ &=\frac {6793 (1+x)}{26180 \sqrt [3]{x^2+x^3}}-\frac {\left (113-23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (113+23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}+\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {209440}{729 (1-x) x^{2/3} \sqrt [3]{1+x}} \, dx}{209440 \sqrt [3]{x^2+x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {209440}{729 x^{2/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{209440 \sqrt [3]{x^2+x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {209440}{729 x^{2/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{209440 \sqrt [3]{x^2+x^3}}\\ &=\frac {6793 (1+x)}{26180 \sqrt [3]{x^2+x^3}}-\frac {\left (113-23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (113+23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}+\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{(1-x) x^{2/3} \sqrt [3]{1+x}} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{x^2+x^3}}\\ &=\frac {6793 (1+x)}{26180 \sqrt [3]{x^2+x^3}}-\frac {\left (113-23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (113+23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}+\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log (1-x)}{6 \sqrt [3]{2} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log \left (1+\sqrt [3]{-1} x\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log \left (1-(-1)^{2/3} x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{2}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-\sqrt [3]{-1}}}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1+(-1)^{2/3}}}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.37, size = 135, normalized size = 0.63 \begin {gather*} \frac {-52360 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x}{x+1}\right )-52360 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x-i \sqrt {3} x}{2 x+2}\right )-52360 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {i \sqrt {3} x+x}{2 x+2}\right )+13473 x^6+4491 x^5-1497 x^4+18285 x^3+900 x^2-660 x+9240}{52360 x^5 \sqrt [3]{x^2 (x+1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9240 - 660*x + 900*x^2 + 18285*x^3 - 1497*x^4 + 4491*x^5 + 13473*x^6 - 52360*x^6*Hypergeometric2F1[1/3, 1, 4/
3, (2*x)/(1 + x)] - 52360*x^6*Hypergeometric2F1[1/3, 1, 4/3, (x - I*Sqrt[3]*x)/(2 + 2*x)] - 52360*x^6*Hypergeo
metric2F1[1/3, 1, 4/3, (x + I*Sqrt[3]*x)/(2 + 2*x)])/(52360*x^5*(x^2*(1 + x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.60, size = 214, normalized size = 1.00 \begin {gather*} \frac {3 \left (x^2+x^3\right )^{2/3} \left (3080-3300 x+3600 x^2+2495 x^3-2994 x^4+4491 x^5\right )}{52360 x^7}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^3}+\sqrt [3]{2} \left (x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(x^2 + x^3)^(2/3)*(3080 - 3300*x + 3600*x^2 + 2495*x^3 - 2994*x^4 + 4491*x^5))/(52360*x^7) - ArcTan[(Sqrt[3
]*x)/(x + 2^(2/3)*(x^2 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) + Log[-2*x + 2^(2/3)*(x^2 + x^3)^(1/3)]/(3*2^(1/3)) -
Log[2*x^2 + 2^(2/3)*x*(x^2 + x^3)^(1/3) + 2^(1/3)*(x^2 + x^3)^(2/3)]/(6*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & ,
(-Log[x] + Log[(x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3

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fricas [B]  time = 0.58, size = 870, normalized size = 4.07

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/157080*(52360*x^7*cos(1/9*pi)*log(16*(x^2 - (2*sqrt(3)*x*cos(1/9*pi)*sin(1/9*pi) + 2*x*cos(1/9*pi)^2 - x)*(x
^3 + x^2)^(1/3) + (x^3 + x^2)^(2/3))/x^2) - 209440*x^7*arctan((8*(2*x*cos(1/9*pi)^3 - x*cos(1/9*pi))*sin(1/9*p
i) + sqrt(3)*x + 2*(2*sqrt(3)*x*cos(1/9*pi)^2 + 2*x*cos(1/9*pi)*sin(1/9*pi) - sqrt(3)*x)*sqrt((x^2 - (2*sqrt(3
)*x*cos(1/9*pi)*sin(1/9*pi) + 2*x*cos(1/9*pi)^2 - x)*(x^3 + x^2)^(1/3) + (x^3 + x^2)^(2/3))/x^2) - 2*(x^3 + x^
2)^(1/3)*(2*sqrt(3)*cos(1/9*pi)^2 + 2*cos(1/9*pi)*sin(1/9*pi) - sqrt(3)))/(16*x*cos(1/9*pi)^4 - 16*x*cos(1/9*p
i)^2 + 3*x))*sin(1/9*pi) + 26180*sqrt(6)*2^(1/6)*x^7*arctan(1/6*2^(1/6)*(sqrt(6)*2^(1/3)*x + 2*sqrt(6)*(x^3 +
x^2)^(1/3))/x) + 26180*2^(2/3)*x^7*log(-(2^(1/3)*x - (x^3 + x^2)^(1/3))/x) - 13090*2^(2/3)*x^7*log((2^(2/3)*x^
2 + 2^(1/3)*(x^3 + x^2)^(1/3)*x + (x^3 + x^2)^(2/3))/x^2) + 104720*(sqrt(3)*x^7*cos(1/9*pi) + x^7*sin(1/9*pi))
*arctan((8*(2*x*cos(1/9*pi)^3 - x*cos(1/9*pi))*sin(1/9*pi) - sqrt(3)*x - 2*(2*sqrt(3)*x*cos(1/9*pi)^2 - 2*x*co
s(1/9*pi)*sin(1/9*pi) - sqrt(3)*x)*sqrt((x^2 + (2*sqrt(3)*x*cos(1/9*pi)*sin(1/9*pi) - 2*x*cos(1/9*pi)^2 + x)*(
x^3 + x^2)^(1/3) + (x^3 + x^2)^(2/3))/x^2) + 2*(x^3 + x^2)^(1/3)*(2*sqrt(3)*cos(1/9*pi)^2 - 2*cos(1/9*pi)*sin(
1/9*pi) - sqrt(3)))/(16*x*cos(1/9*pi)^4 - 16*x*cos(1/9*pi)^2 + 3*x)) - 104720*(sqrt(3)*x^7*cos(1/9*pi) - x^7*s
in(1/9*pi))*arctan(-1/2*(2*x*cos(1/9*pi)^2 - x*sqrt((x^2 + 2*(x^3 + x^2)^(1/3)*(2*x*cos(1/9*pi)^2 - x) + (x^3
+ x^2)^(2/3))/x^2) - x + (x^3 + x^2)^(1/3))/(x*cos(1/9*pi)*sin(1/9*pi))) + 26180*(sqrt(3)*x^7*sin(1/9*pi) - x^
7*cos(1/9*pi))*log(64*(x^2 + (2*sqrt(3)*x*cos(1/9*pi)*sin(1/9*pi) - 2*x*cos(1/9*pi)^2 + x)*(x^3 + x^2)^(1/3) +
(x^3 + x^2)^(2/3))/x^2) - 26180*(sqrt(3)*x^7*sin(1/9*pi) + x^7*cos(1/9*pi))*log(64*(x^2 + 2*(x^3 + x^2)^(1/3)
*(2*x*cos(1/9*pi)^2 - x) + (x^3 + x^2)^(2/3))/x^2) + 9*(4491*x^5 - 2994*x^4 + 2495*x^3 + 3600*x^2 - 3300*x + 3
080)*(x^3 + x^2)^(2/3))/x^7

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giac [B]  time = 68.49, size = 986, normalized size = 4.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/17*(1/x + 1)^(17/3) - 15/14*(1/x + 1)^(14/3) + 30/11*(1/x + 1)^(11/3) - 27/8*(1/x + 1)^(8/3) + 1/6*sqrt(3)*2
^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(1/x + 1)^(1/3))) - 1/3*(sqrt(3)*cos(4/9*pi)^5 - 10*sqrt(3)*cos
(4/9*pi)^3*sin(4/9*pi)^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 5*cos(4/9*pi)^4*sin(4/9*pi) + 10*cos(4/9*pi)^
2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3)*cos(4/9*pi)^2 - sqrt(3)*sin(4/9*pi)^2 - 2*cos(4/9*pi)*sin(4/9*pi))*a
rctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi))) - 1/3*(sqrt(
3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^4 - 5*cos(2/9*pi
)^4*sin(2/9*pi) + 10*cos(2/9*pi)^2*sin(2/9*pi)^3 - sin(2/9*pi)^5 + sqrt(3)*cos(2/9*pi)^2 - sqrt(3)*sin(2/9*pi)
^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(2/9*pi) + 2*(1/x + 1)^(1/3))/((1/2*I*sqrt(3)
+ 1/2)*sin(2/9*pi))) + 1/3*(sqrt(3)*cos(1/9*pi)^5 - 10*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi)^2 + 5*sqrt(3)*cos(1/9
*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*pi) - 10*cos(1/9*pi)^2*sin(1/9*pi)^3 + sin(1/9*pi)^5 - sqrt(3)*co
s(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi))*arctan(-1/2*((-I*sqrt(3) - 1)*cos(1/9*pi) - 2
*(1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) - 1/6*(5*sqrt(3)*cos(4/9*pi)^4*sin(4/9*pi) - 10*sqrt(3)
*cos(4/9*pi)^2*sin(4/9*pi)^3 + sqrt(3)*sin(4/9*pi)^5 + cos(4/9*pi)^5 - 10*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*cos(
4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) + cos(4/9*pi)^2 - sin(4/9*pi)^2)*log((-I*sqrt(3)*cos
(4/9*pi) - cos(4/9*pi))*(1/x + 1)^(1/3) + (1/x + 1)^(2/3) + 1) - 1/6*(5*sqrt(3)*cos(2/9*pi)^4*sin(2/9*pi) - 10
*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^3 + sqrt(3)*sin(2/9*pi)^5 + cos(2/9*pi)^5 - 10*cos(2/9*pi)^3*sin(2/9*pi)^2
+ 5*cos(2/9*pi)*sin(2/9*pi)^4 + 2*sqrt(3)*cos(2/9*pi)*sin(2/9*pi) + cos(2/9*pi)^2 - sin(2/9*pi)^2)*log((-I*sqr
t(3)*cos(2/9*pi) - cos(2/9*pi))*(1/x + 1)^(1/3) + (1/x + 1)^(2/3) + 1) - 1/6*(5*sqrt(3)*cos(1/9*pi)^4*sin(1/9*
pi) - 10*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^3 + sqrt(3)*sin(1/9*pi)^5 - cos(1/9*pi)^5 + 10*cos(1/9*pi)^3*sin(1/
9*pi)^2 - 5*cos(1/9*pi)*sin(1/9*pi)^4 - 2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) + cos(1/9*pi)^2 - sin(1/9*pi)^2)*log
((I*sqrt(3)*cos(1/9*pi) + cos(1/9*pi))*(1/x + 1)^(1/3) + (1/x + 1)^(2/3) + 1) + 9/5*(1/x + 1)^(5/3) - 1/12*2^(
2/3)*log(2^(2/3) + 2^(1/3)*(1/x + 1)^(1/3) + (1/x + 1)^(2/3)) + 1/6*2^(2/3)*log(abs(-2^(1/3) + (1/x + 1)^(1/3)
))

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maple [B]  time = 13.09, size = 1916, normalized size = 8.95

method	result	size
risch	\(\text {Expression too large to display}\)	\(1916\)
trager	\(\text {Expression too large to display}\)	\(2268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/52360*(4491*x^6+1497*x^5-499*x^4+6095*x^3+300*x^2-220*x+3080)/x^5/(x^2*(1+x))^(1/3)-81/32*ln(-(2106*RootOf(7
29*_Z^6-216*_Z^3+64)^5*x^2-1296*RootOf(729*_Z^6-216*_Z^3+64)^4*(x^3+x^2)^(1/3)*x+1053*RootOf(729*_Z^6-216*_Z^3
+64)^5*x+324*(x^3+x^2)^(2/3)*RootOf(729*_Z^6-216*_Z^3+64)^3-528*RootOf(729*_Z^6-216*_Z^3+64)^2*x^2+144*RootOf(
729*_Z^6-216*_Z^3+64)*(x^3+x^2)^(1/3)*x-264*x*RootOf(729*_Z^6-216*_Z^3+64)^2+160*(x^3+x^2)^(2/3))/(27*RootOf(7
29*_Z^6-216*_Z^3+64)^3*x-54*RootOf(729*_Z^6-216*_Z^3+64)^3-24*x-8)/x)*RootOf(729*_Z^6-216*_Z^3+64)^5+3/4*ln(-(
2106*RootOf(729*_Z^6-216*_Z^3+64)^5*x^2-1296*RootOf(729*_Z^6-216*_Z^3+64)^4*(x^3+x^2)^(1/3)*x+1053*RootOf(729*
_Z^6-216*_Z^3+64)^5*x+324*(x^3+x^2)^(2/3)*RootOf(729*_Z^6-216*_Z^3+64)^3-528*RootOf(729*_Z^6-216*_Z^3+64)^2*x^
2+144*RootOf(729*_Z^6-216*_Z^3+64)*(x^3+x^2)^(1/3)*x-264*x*RootOf(729*_Z^6-216*_Z^3+64)^2+160*(x^3+x^2)^(2/3))
/(27*RootOf(729*_Z^6-216*_Z^3+64)^3*x-54*RootOf(729*_Z^6-216*_Z^3+64)^3-24*x-8)/x)*RootOf(729*_Z^6-216*_Z^3+64
)^2-27/16*RootOf(729*_Z^6-216*_Z^3+64)^4*ln((-729*RootOf(729*_Z^6-216*_Z^3+64)^5*(x^3+x^2)^(1/3)*x-1404*RootOf
(729*_Z^6-216*_Z^3+64)^4*x^2+324*(x^3+x^2)^(2/3)*RootOf(729*_Z^6-216*_Z^3+64)^3-702*RootOf(729*_Z^6-216*_Z^3+6
4)^4*x+576*RootOf(729*_Z^6-216*_Z^3+64)^2*(x^3+x^2)^(1/3)*x+64*x^2*RootOf(729*_Z^6-216*_Z^3+64)-256*(x^3+x^2)^
(2/3)+32*RootOf(729*_Z^6-216*_Z^3+64)*x)/(27*RootOf(729*_Z^6-216*_Z^3+64)^3*x-54*RootOf(729*_Z^6-216*_Z^3+64)^
3+16*x+24)/x)+1/2*RootOf(729*_Z^6-216*_Z^3+64)*ln(-(-729*RootOf(729*_Z^6-216*_Z^3+64)^6*x^2+1458*RootOf(729*_Z
^6-216*_Z^3+64)^6*x-6804*RootOf(729*_Z^6-216*_Z^3+64)^4*(x^3+x^2)^(1/3)*x+1728*RootOf(729*_Z^6-216*_Z^3+64)^3*
x^2+3024*(x^3+x^2)^(2/3)*RootOf(729*_Z^6-216*_Z^3+64)^2-1944*RootOf(729*_Z^6-216*_Z^3+64)^3*x-960*x^2-320*x)/(
27*RootOf(729*_Z^6-216*_Z^3+64)^3*x-54*RootOf(729*_Z^6-216*_Z^3+64)^3+16*x+24)/x)-3/4*RootOf(729*_Z^6-216*_Z^3
+64)^2*ln((-2187*RootOf(729*_Z^6-216*_Z^3+64)^8*x^2+4374*RootOf(729*_Z^6-216*_Z^3+64)^8*x-3888*RootOf(729*_Z^6
-216*_Z^3+64)^5*x^2+3240*RootOf(729*_Z^6-216*_Z^3+64)^5*x+6048*(x^3+x^2)^(2/3)*RootOf(729*_Z^6-216*_Z^3+64)^3-
1536*RootOf(729*_Z^6-216*_Z^3+64)^2*x^2+2688*RootOf(729*_Z^6-216*_Z^3+64)*(x^3+x^2)^(1/3)*x-2304*x*RootOf(729*
_Z^6-216*_Z^3+64)^2-1792*(x^3+x^2)^(2/3))/(27*RootOf(729*_Z^6-216*_Z^3+64)^3*x-54*RootOf(729*_Z^6-216*_Z^3+64)
^3-24*x-8)/x)+1/6*RootOf(_Z^3-4)*ln(-(30*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*
x^2-9*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x^2-60*RootOf(RootOf(_Z^3-4)^2+3*
_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x+360*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9
*_Z^2)*RootOf(_Z^3-4)^2+18*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x-240*(x^3+x
^2)^(1/3)*RootOf(_Z^3-4)^2*x-594*(x^3+x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z
^3-4)*x+340*RootOf(_Z^3-4)*x^2-102*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x^2+60*RootOf(_Z^3-4)*x
+84*(x^3+x^2)^(2/3)-18*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x)/x/(-1+x))+1/2*RootOf(RootOf(_Z^3
-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*ln(-(3*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*
x^2-90*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x^2-6*RootOf(RootOf(_Z^3-4)^2+3*
_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x-360*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9
*_Z^2)*RootOf(_Z^3-4)^2+180*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x+240*(x^3+
x^2)^(1/3)*RootOf(_Z^3-4)^2*x+126*(x^3+x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_
Z^3-4)*x-30*RootOf(_Z^3-4)*x^2+900*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x^2-14*RootOf(_Z^3-4)*x
-396*(x^3+x^2)^(2/3)+420*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x)/x/(-1+x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^6-x^9\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-int(1/((x^2 + x^3)^(1/3)*(x^6 - x^9)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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