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

Optimal. Leaf size=101 \[ \frac {\left (x^3+x^2\right )^{2/3} \left (109573 x^6+19071 x^5-6357 x^4+20985 x^3-900 x^2+660 x-9240\right )}{52360 x^7 (x+1)}-\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-3 \text {$\#$1}^3+3\& ,\frac {\log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]

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Rubi [C]  time = 1.16, antiderivative size = 844, normalized size of antiderivative = 8.36, number of steps used = 27, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2056, 6725, 21, 45, 37, 129, 155, 12, 91} \begin {gather*} -\frac {\left (1511+4777 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}-\frac {2187 (x+1)}{1309 x \sqrt [3]{x^3+x^2}}+\frac {\left (2249+153 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (2249-153 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {3645 (x+1)}{2618 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (41+17 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}+\frac {\left (41-17 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}-\frac {1620 (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 {135 (x+1)}{119 x^4 \sqrt [3]{x^3+x^2}}-\frac {20 (x+1)}{17 x^5 \sqrt [3]{x^3+x^2}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}+\frac {6561 (x+1)}{2618 \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 \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)^{2/3} x-1\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]{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}}+\frac {1}{x^5 \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]

1/(x^5*(x^2 + x^3)^(1/3)) + (6561*(1 + x))/(2618*(x^2 + x^3)^(1/3)) - ((21647 - (11849*I)*Sqrt[3])*(1 + x))/(1
04720*(x^2 + x^3)^(1/3)) - ((21647 + (11849*I)*Sqrt[3])*(1 + x))/(104720*(x^2 + x^3)^(1/3)) - (20*(1 + x))/(17
*x^5*(x^2 + x^3)^(1/3)) + (135*(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)) - (1620*(1 + x))/(1309*x^3*(x^
2 + x^3)^(1/3)) + ((41 - (17*I)*Sqrt[3])*(1 + x))/(2618*x^3*(x^2 + x^3)^(1/3)) + ((41 + (17*I)*Sqrt[3])*(1 + x
))/(2618*x^3*(x^2 + x^3)^(1/3)) + (3645*(1 + x))/(2618*x^2*(x^2 + x^3)^(1/3)) + ((2249 - (153*I)*Sqrt[3])*(1 +
x))/(20944*x^2*(x^2 + x^3)^(1/3)) + ((2249 + (153*I)*Sqrt[3])*(1 + x))/(20944*x^2*(x^2 + x^3)^(1/3)) - (2187*
(1 + x))/(1309*x*(x^2 + x^3)^(1/3)) - ((1511 - (4777*I)*Sqrt[3])*(1 + x))/(52360*x*(x^2 + x^3)^(1/3)) - ((1511
+ (4777*I)*Sqrt[3])*(1 + x))/(52360*x*(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)*ArcTan[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 + (-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)/(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 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 37

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] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

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*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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 {2 (1+x)}{17 x^5 \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 {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} (1+x)^{4/3}} \, dx}{3 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {2 (1+x)}{17 x^5 \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 {1}{9} \left (-41+17 i \sqrt {3}\right )-\frac {4}{3} \left (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 (-41-17 i \sqrt {3}\right )-\frac {4}{3} \left (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 {\left (6 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x}} \, dx}{\sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \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 (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 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 {1}{27} \left (-2249+153 i \sqrt {3}\right )-\frac {2}{3} \left (23-6 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 {\left (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{27} \left (-2249-153 i \sqrt {3}\right )-\frac {2}{3} \left (23+6 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 (90 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{17/3} \sqrt [3]{1+x}} \, dx}{17 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (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 (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 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 {2}{81} \left (1511+4777 i \sqrt {3}\right )-\frac {2}{27} \left (895-1201 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 {\left (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{81} \left (1511-4777 i \sqrt {3}\right )-\frac {2}{27} \left (895+1201 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 (540 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{14/3} \sqrt [3]{1+x}} \, dx}{119 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (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 {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 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 {2}{243} \left (21647-11849 i \sqrt {3}\right )+\frac {2}{81} \left (7921-1633 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 {\left (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{243} \left (21647+11849 i \sqrt {3}\right )+\frac {2}{81} \left (7921+1633 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 (4860 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{1+x}} \, dx}{1309 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (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 {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {3645 (1+x)}{2618 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 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 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 {\left (3645 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{8/3} \sqrt [3]{1+x}} \, dx}{1309 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (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 {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {3645 (1+x)}{2618 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {2187 (1+x)}{1309 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 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}{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 {\left (2187 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{5/3} \sqrt [3]{1+x}} \, dx}{1309 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}+\frac {6561 (1+x)}{2618 \sqrt [3]{x^2+x^3}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (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 {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {3645 (1+x)}{2618 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {2187 (1+x)}{1309 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 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 \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 \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]{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 = 0.32, size = 116, normalized size = 1.15 \begin {gather*} \frac {52360 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (3-i \sqrt {3}\right ) x}{2 (x+1)}\right )+52360 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (3+i \sqrt {3}\right ) x}{2 (x+1)}\right )+109573 x^6+19071 x^5-6357 x^4+20985 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 + 20985*x^3 - 6357*x^4 + 19071*x^5 + 109573*x^6 + 52360*x^6*Hypergeometric2F1[1/3, 1,
4/3, ((3 - I*Sqrt[3])*x)/(2*(1 + x))] + 52360*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((3 + I*Sqrt[3])*x)/(2*(1 +
x))])/(52360*x^5*(x^2*(1 + x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.35, size = 101, normalized size = 1.00 \begin {gather*} \frac {\left (x^2+x^3\right )^{2/3} \left (-9240+660 x-900 x^2+20985 x^3-6357 x^4+19071 x^5+109573 x^6\right )}{52360 x^7 (1+x)}-\frac {1}{3} \text {RootSum}\left [3-3 \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]

((x^2 + x^3)^(2/3)*(-9240 + 660*x - 900*x^2 + 20985*x^3 - 6357*x^4 + 19071*x^5 + 109573*x^6))/(52360*x^7*(1 +
x)) - RootSum[3 - 3*#1^3 + #1^6 & , (-Log[x] + Log[(x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3

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fricas [B]  time = 0.65, size = 1412, normalized size = 13.98

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]

1/471240*(26180*12^(1/6)*6^(2/3)*(x^8 + x^7)*cos(2/3*arctan(sqrt(3) + 2))*log(12*(4*12^(1/3)*6^(1/3)*sqrt(3)*(
x^3 + x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) - 12*12^(1/3)*6^(1/3)*(x^3 + x^2)
^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 + 12^(2/3)*6^(2/3)*x^2 + 6*12^(1/3)*6^(1/3)*(x^3 + x^2)^(1/3)*x + 12*(
x^3 + x^2)^(2/3))/x^2) - 104720*12^(1/6)*6^(2/3)*(x^8 + x^7)*arctan(1/108*(12*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 +
x^2)^(1/3)*cos(2/3*arctan(sqrt(3) + 2))^2 - 6*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 + x^2)^(1/3) - sqrt(3)*(2*12^(2/3)
*6^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 - 6*12^(2/3)*6^(2/3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*
arctan(sqrt(3) + 2)) - 12^(2/3)*6^(2/3)*sqrt(3)*x)*sqrt((4*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 + x^2)^(1/3)*x*cos(2/
3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) - 12*12^(1/3)*6^(1/3)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(s
qrt(3) + 2))^2 + 12^(2/3)*6^(2/3)*x^2 + 6*12^(1/3)*6^(1/3)*(x^3 + x^2)^(1/3)*x + 12*(x^3 + x^2)^(2/3))/x^2) +
36*(48*x*cos(2/3*arctan(sqrt(3) + 2))^3 - (12^(2/3)*6^(2/3)*(x^3 + x^2)^(1/3) + 24*x)*cos(2/3*arctan(sqrt(3) +
2)))*sin(2/3*arctan(sqrt(3) + 2)) - 108*sqrt(3)*x)/(16*x*cos(2/3*arctan(sqrt(3) + 2))^4 - 16*x*cos(2/3*arctan
(sqrt(3) + 2))^2 + x))*sin(2/3*arctan(sqrt(3) + 2)) - 52360*(12^(1/6)*6^(2/3)*sqrt(3)*(x^8 + x^7)*cos(2/3*arct
an(sqrt(3) + 2)) + 12^(1/6)*6^(2/3)*(x^8 + x^7)*sin(2/3*arctan(sqrt(3) + 2)))*arctan(-1/108*(12*12^(2/3)*6^(2/
3)*sqrt(3)*(x^3 + x^2)^(1/3)*cos(2/3*arctan(sqrt(3) + 2))^2 - 6*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 + x^2)^(1/3) - s
qrt(3)*(2*12^(2/3)*6^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 + 6*12^(2/3)*6^(2/3)*x*cos(2/3*arctan(sqrt
(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) - 12^(2/3)*6^(2/3)*sqrt(3)*x)*sqrt((4*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 + x
^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) + 12*12^(1/3)*6^(1/3)*(x^3 + x^2)^(1/3)*
x*cos(2/3*arctan(sqrt(3) + 2))^2 + 12^(2/3)*6^(2/3)*x^2 - 6*12^(1/3)*6^(1/3)*(x^3 + x^2)^(1/3)*x + 12*(x^3 + x
^2)^(2/3))/x^2) + 36*(48*x*cos(2/3*arctan(sqrt(3) + 2))^3 + (12^(2/3)*6^(2/3)*(x^3 + x^2)^(1/3) - 24*x)*cos(2/
3*arctan(sqrt(3) + 2)))*sin(2/3*arctan(sqrt(3) + 2)) + 108*sqrt(3)*x)/(16*x*cos(2/3*arctan(sqrt(3) + 2))^4 - 1
6*x*cos(2/3*arctan(sqrt(3) + 2))^2 + x)) - 52360*(12^(1/6)*6^(2/3)*sqrt(3)*(x^8 + x^7)*cos(2/3*arctan(sqrt(3)
+ 2)) - 12^(1/6)*6^(2/3)*(x^8 + x^7)*sin(2/3*arctan(sqrt(3) + 2)))*arctan(1/72*(144*x*cos(2/3*arctan(sqrt(3) +
2))*sin(2/3*arctan(sqrt(3) + 2)) + 12^(2/3)*6^(2/3)*x*sqrt(-(8*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 + x^2)^(1/3)*x*c
os(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) - 12^(2/3)*6^(2/3)*x^2 - 12*(x^3 + x^2)^(2/3))/x^2) -
2*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 + x^2)^(1/3))/(2*x*cos(2/3*arctan(sqrt(3) + 2))^2 - x)) - 13090*(12^(1/6)*6^(
2/3)*sqrt(3)*(x^8 + x^7)*sin(2/3*arctan(sqrt(3) + 2)) + 12^(1/6)*6^(2/3)*(x^8 + x^7)*cos(2/3*arctan(sqrt(3) +
2)))*log(-48*(8*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(
3) + 2)) - 12^(2/3)*6^(2/3)*x^2 - 12*(x^3 + x^2)^(2/3))/x^2) + 13090*(12^(1/6)*6^(2/3)*sqrt(3)*(x^8 + x^7)*sin
(2/3*arctan(sqrt(3) + 2)) - 12^(1/6)*6^(2/3)*(x^8 + x^7)*cos(2/3*arctan(sqrt(3) + 2)))*log(48*(4*12^(1/3)*6^(1
/3)*sqrt(3)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) + 12*12^(1/3)*6^(1/3
)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 + 12^(2/3)*6^(2/3)*x^2 - 6*12^(1/3)*6^(1/3)*(x^3 + x^2)^(
1/3)*x + 12*(x^3 + x^2)^(2/3))/x^2) + 9*(109573*x^6 + 19071*x^5 - 6357*x^4 + 20985*x^3 - 900*x^2 + 660*x - 924
0)*(x^3 + x^2)^(2/3))/(x^8 + x^7)

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giac [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="giac")

[Out]

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

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maple [B]  time = 8.67, size = 2059, normalized size = 20.39

method	result	size
risch	\(\text {Expression too large to display}\)	\(2059\)
trager	\(\text {Expression too large to display}\)	\(2821\)

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]

1/52360*(109573*x^6+19071*x^5-6357*x^4+20985*x^3-900*x^2+660*x-9240)/x^5/(x^2*(1+x))^(1/3)+1/3*RootOf(RootOf(3
*_Z^6+3*_Z^3+1)^3+_Z^3+1)*ln(-(-6*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x^2+6*Roo
tOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x-39*(x^3+x^2)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z
^3+1)^3+_Z^3+1)^2*RootOf(3*_Z^6+3*_Z^3+1)^3*x-10*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3
+1)^3*x^2+24*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3+17*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z
^6+3*_Z^3+1)^3*x-15*(x^3+x^2)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*x+4*RootOf(RootOf(3*_Z^6+3*_Z^3
+1)^3+_Z^3+1)*x^2+13*(x^3+x^2)^(2/3)+10*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x)/(3*x*RootOf(3*_Z^6+3*_Z^3+
1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^3+2*x-1)/x)-RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*
ln((21*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x^2-21*RootOf(RootOf(3*_Z^6+3*_Z^3+1
)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x+3*(x^3+x^2)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*RootOf(3*
_Z^6+3*_Z^3+1)^3*x+89*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x^2+21*(x^3+x^2)^(2/3
)*RootOf(3*_Z^6+3*_Z^3+1)^3+2*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x+24*(x^3+x^2
)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*x+44*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x^2-(x^3+x^2)
^(2/3)+8*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x)/(3*x*RootOf(3*_Z^6+3*_Z^3+1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^
3+2*x-1)/x)-2/3*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*ln((21*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootO
f(3*_Z^6+3*_Z^3+1)^6*x^2-21*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x+3*(x^3+x^2)^(
1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*RootOf(3*_Z^6+3*_Z^3+1)^3*x+89*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^
3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x^2+21*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3+2*RootOf(RootOf(3*_Z^6+3*
_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x+24*(x^3+x^2)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*x+
44*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x^2-(x^3+x^2)^(2/3)+8*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x)/
(3*x*RootOf(3*_Z^6+3*_Z^3+1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^3+2*x-1)/x)+1/3*RootOf(3*_Z^6+3*_Z^3+1)*ln((6*RootOf(
3*_Z^6+3*_Z^3+1)^7*x^2-6*RootOf(3*_Z^6+3*_Z^3+1)^7*x-39*(x^3+x^2)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^5*x+2*RootOf(3
*_Z^6+3*_Z^3+1)^4*x^2+24*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3+5*RootOf(3*_Z^6+3*_Z^3+1)^4*x-24*(x^3+x^2)^
(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^2*x-8*RootOf(3*_Z^6+3*_Z^3+1)*x^2+11*(x^3+x^2)^(2/3)+RootOf(3*_Z^6+3*_Z^3+1)*x)/
(3*x*RootOf(3*_Z^6+3*_Z^3+1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^3+x-2)/x)+RootOf(3*_Z^6+3*_Z^3+1)^4*ln(-(-21*RootOf(3
*_Z^6+3*_Z^3+1)^7*x^2+21*RootOf(3*_Z^6+3*_Z^3+1)^7*x+3*(x^3+x^2)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^5*x+47*RootOf(3
*_Z^6+3*_Z^3+1)^4*x^2+21*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3+44*RootOf(3*_Z^6+3*_Z^3+1)^4*x-21*(x^3+x^2)
^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^2*x+24*RootOf(3*_Z^6+3*_Z^3+1)*x^2+22*(x^3+x^2)^(2/3)+15*RootOf(3*_Z^6+3*_Z^3+1
)*x)/(3*x*RootOf(3*_Z^6+3*_Z^3+1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^3+x-2)/x)+1/3*RootOf(3*_Z^6+3*_Z^3+1)*ln(-(-21*R
ootOf(3*_Z^6+3*_Z^3+1)^7*x^2+21*RootOf(3*_Z^6+3*_Z^3+1)^7*x+3*(x^3+x^2)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^5*x+47*R
ootOf(3*_Z^6+3*_Z^3+1)^4*x^2+21*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3+44*RootOf(3*_Z^6+3*_Z^3+1)^4*x-21*(x
^3+x^2)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^2*x+24*RootOf(3*_Z^6+3*_Z^3+1)*x^2+22*(x^3+x^2)^(2/3)+15*RootOf(3*_Z^6+3
*_Z^3+1)*x)/(3*x*RootOf(3*_Z^6+3*_Z^3+1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^3+x-2)/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 = -0.01 \begin {gather*} \int \frac {1}{x^6\,{\left (x^3+x^2\right )}^{1/3}\,\left (x^3+1\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^6*(x^2 + x^3)^(1/3)*(x^3 + 1)), 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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