3.17.57 \(\int \frac {\sqrt [3]{-6-2 x+x^2}}{-1+x} \, dx\)
Optimal. Leaf size=141 \[ \frac {3}{2} \sqrt [3]{x^2-2 x-6}-\frac {1}{2} \sqrt [3]{7} \log \left (7^{2/3} \sqrt [3]{x^2-2 x-6}+7\right )+\frac {1}{4} \sqrt [3]{7} \log \left (-\sqrt [3]{7} \left (x^2-2 x-6\right )^{2/3}+7^{2/3} \sqrt [3]{x^2-2 x-6}-7\right )+\frac {1}{2} \sqrt {3} \sqrt [3]{7} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^2-2 x-6}}{\sqrt {3} \sqrt [3]{7}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 103, normalized size of antiderivative = 0.73,
number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used =
{694, 266, 50, 58, 617, 204, 31} \begin {gather*} \frac {3}{2} \sqrt [3]{(x-1)^2-7}-\frac {3}{4} \sqrt [3]{7} \log \left (\sqrt [3]{(x-1)^2-7}+\sqrt [3]{7}\right )+\frac {1}{2} \sqrt [3]{7} \log (1-x)+\frac {1}{2} \sqrt {3} \sqrt [3]{7} \tan ^{-1}\left (\frac {\sqrt [3]{7}-2 \sqrt [3]{(x-1)^2-7}}{\sqrt {3} \sqrt [3]{7}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-6 - 2*x + x^2)^(1/3)/(-1 + x),x]
[Out]
(3*(-7 + (-1 + x)^2)^(1/3))/2 + (Sqrt[3]*7^(1/3)*ArcTan[(7^(1/3) - 2*(-7 + (-1 + x)^2)^(1/3))/(Sqrt[3]*7^(1/3)
)])/2 - (3*7^(1/3)*Log[7^(1/3) + (-7 + (-1 + x)^2)^(1/3)])/4 + (7^(1/3)*Log[1 - x])/2
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 58
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d
*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x
] && NegQ[(b*c - a*d)/b]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[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 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 694
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[x^m*(
a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-6-2 x+x^2}}{-1+x} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt [3]{-7+x^2}}{x} \, dx,x,-1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-7+x}}{x} \, dx,x,(-1+x)^2\right )\\ &=\frac {3}{2} \sqrt [3]{-7+(-1+x)^2}-\frac {7}{2} \operatorname {Subst}\left (\int \frac {1}{(-7+x)^{2/3} x} \, dx,x,(-1+x)^2\right )\\ &=\frac {3}{2} \sqrt [3]{-7+(-1+x)^2}+\frac {1}{2} \sqrt [3]{7} \log (1-x)-\frac {1}{4} \left (3 \sqrt [3]{7}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{7}+x} \, dx,x,\sqrt [3]{-7+(-1+x)^2}\right )-\frac {1}{4} \left (3\ 7^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{7^{2/3}-\sqrt [3]{7} x+x^2} \, dx,x,\sqrt [3]{-7+(-1+x)^2}\right )\\ &=\frac {3}{2} \sqrt [3]{-7+(-1+x)^2}-\frac {3}{4} \sqrt [3]{7} \log \left (\sqrt [3]{7}+\sqrt [3]{-7+(-1+x)^2}\right )+\frac {1}{2} \sqrt [3]{7} \log (1-x)-\frac {1}{2} \left (3 \sqrt [3]{7}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-7+(-1+x)^2}}{\sqrt [3]{7}}\right )\\ &=\frac {3}{2} \sqrt [3]{-7+(-1+x)^2}+\frac {1}{2} \sqrt {3} \sqrt [3]{7} \tan ^{-1}\left (\frac {7-2\ 7^{2/3} \sqrt [3]{-7+(-1+x)^2}}{7 \sqrt {3}}\right )-\frac {3}{4} \sqrt [3]{7} \log \left (\sqrt [3]{7}+\sqrt [3]{-7+(-1+x)^2}\right )+\frac {1}{2} \sqrt [3]{7} \log (1-x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 129, normalized size = 0.91 \begin {gather*} \frac {1}{4} \left (6 \sqrt [3]{x^2-2 x-6}+2 \sqrt {3} \sqrt [3]{7} \tan ^{-1}\left (\frac {7-2\ 7^{2/3} \sqrt [3]{x^2-2 x-6}}{7 \sqrt {3}}\right )-2 \sqrt [3]{7} \log \left (\sqrt [3]{(x-1)^2-7}+\sqrt [3]{7}\right )+\sqrt [3]{7} \log \left (\left ((x-1)^2-7\right )^{2/3}-\sqrt [3]{7} \sqrt [3]{(x-1)^2-7}+7^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-6 - 2*x + x^2)^(1/3)/(-1 + x),x]
[Out]
(6*(-6 - 2*x + x^2)^(1/3) + 2*Sqrt[3]*7^(1/3)*ArcTan[(7 - 2*7^(2/3)*(-6 - 2*x + x^2)^(1/3))/(7*Sqrt[3])] - 2*7
^(1/3)*Log[7^(1/3) + (-7 + (-1 + x)^2)^(1/3)] + 7^(1/3)*Log[7^(2/3) - 7^(1/3)*(-7 + (-1 + x)^2)^(1/3) + (-7 +
(-1 + x)^2)^(2/3)])/4
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.31, size = 141, normalized size = 1.00 \begin {gather*} \frac {3}{2} \sqrt [3]{x^2-2 x-6}-\frac {1}{2} \sqrt [3]{7} \log \left (7^{2/3} \sqrt [3]{x^2-2 x-6}+7\right )+\frac {1}{4} \sqrt [3]{7} \log \left (-\sqrt [3]{7} \left (x^2-2 x-6\right )^{2/3}+7^{2/3} \sqrt [3]{x^2-2 x-6}-7\right )+\frac {1}{2} \sqrt {3} \sqrt [3]{7} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^2-2 x-6}}{\sqrt {3} \sqrt [3]{7}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(-6 - 2*x + x^2)^(1/3)/(-1 + x),x]
[Out]
(3*(-6 - 2*x + x^2)^(1/3))/2 + (Sqrt[3]*7^(1/3)*ArcTan[1/Sqrt[3] - (2*(-6 - 2*x + x^2)^(1/3))/(Sqrt[3]*7^(1/3)
)])/2 - (7^(1/3)*Log[7 + 7^(2/3)*(-6 - 2*x + x^2)^(1/3)])/2 + (7^(1/3)*Log[-7 + 7^(2/3)*(-6 - 2*x + x^2)^(1/3)
- 7^(1/3)*(-6 - 2*x + x^2)^(2/3)])/4
________________________________________________________________________________________
fricas [A] time = 0.39, size = 102, normalized size = 0.72 \begin {gather*} \frac {1}{2} \, \sqrt {3} \left (-7\right )^{\frac {1}{3}} \arctan \left (\frac {2}{21} \, \sqrt {3} \left (-7\right )^{\frac {2}{3}} {\left (x^{2} - 2 \, x - 6\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{4} \, \left (-7\right )^{\frac {1}{3}} \log \left (\left (-7\right )^{\frac {2}{3}} + \left (-7\right )^{\frac {1}{3}} {\left (x^{2} - 2 \, x - 6\right )}^{\frac {1}{3}} + {\left (x^{2} - 2 \, x - 6\right )}^{\frac {2}{3}}\right ) + \frac {1}{2} \, \left (-7\right )^{\frac {1}{3}} \log \left (-\left (-7\right )^{\frac {1}{3}} + {\left (x^{2} - 2 \, x - 6\right )}^{\frac {1}{3}}\right ) + \frac {3}{2} \, {\left (x^{2} - 2 \, x - 6\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-2*x-6)^(1/3)/(-1+x),x, algorithm="fricas")
[Out]
1/2*sqrt(3)*(-7)^(1/3)*arctan(2/21*sqrt(3)*(-7)^(2/3)*(x^2 - 2*x - 6)^(1/3) - 1/3*sqrt(3)) - 1/4*(-7)^(1/3)*lo
g((-7)^(2/3) + (-7)^(1/3)*(x^2 - 2*x - 6)^(1/3) + (x^2 - 2*x - 6)^(2/3)) + 1/2*(-7)^(1/3)*log(-(-7)^(1/3) + (x
^2 - 2*x - 6)^(1/3)) + 3/2*(x^2 - 2*x - 6)^(1/3)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} - 2 \, x - 6\right )}^{\frac {1}{3}}}{x - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-2*x-6)^(1/3)/(-1+x),x, algorithm="giac")
[Out]
integrate((x^2 - 2*x - 6)^(1/3)/(x - 1), x)
________________________________________________________________________________________
maple [C] time = 8.15, size = 2698, normalized size = 19.13 \begin {gather*} \text {Expression too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2-2*x-6)^(1/3)/(-1+x),x)
[Out]
3/2*(x^2-2*x-6)^(1/3)+(1/2*RootOf(_Z^3+7)*ln((-8424*RootOf(_Z^3+7)^2*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)+16758*(x^
4-4*x^3-8*x^2+24*x+36)^(1/3)*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)*x^2+10054
8*(x^4-4*x^3-8*x^2+24*x+36)^(2/3)-252*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*x^4+1008*RootOf
(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*x^3-32004*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+4
9*_Z^2)+521208*RootOf(_Z^3+7)+5838*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*x^2-13692*RootOf(3
6*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*x+4104*RootOf(_Z^3+7)*x^4-16416*RootOf(_Z^3+7)*x^3-95076*Root
Of(_Z^3+7)*x^2+222984*RootOf(_Z^3+7)*x+798*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^
3+7)^3*x^4-3192*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)^3*x^3-33516*(x^4-4*x^3
-8*x^2+24*x+36)^(1/3)*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)*x-49*RootOf(36*R
ootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)^2*RootOf(_Z^3+7)^2*x^4+196*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*Root
Of(_Z^3+7)+49*_Z^2)^2*RootOf(_Z^3+7)^2*x^3+98*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)^2*RootO
f(_Z^3+7)^2*x^2-1596*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)^3*x^2-588*RootOf(
36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)^2*RootOf(_Z^3+7)^2*x+9576*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*R
ootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)^3*x-15120*(x^4-4*x^3-8*x^2+24*x+36)^(2/3)*RootOf(36*RootOf(_Z^3+7)^2+42*
_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)^2+1404*RootOf(_Z^3+7)^2*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)*x^2-2808*Roo
tOf(_Z^3+7)^2*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)*x-100548*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)*RootOf(36*RootOf(_Z^3+7
)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7))/(x^2-2*x-6)/(-1+x)^2)-1/2*ln((-43092*RootOf(_Z^3+7)^2*(x^4-4
*x^3-8*x^2+24*x+36)^(1/3)+819*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+
49*_Z^2)*RootOf(_Z^3+7)*x^2+4914*(x^4-4*x^3-8*x^2+24*x+36)^(2/3)-5460*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(
_Z^3+7)+49*_Z^2)*x^4+21840*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*x^3-320040*RootOf(36*RootO
f(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)-260604*RootOf(_Z^3+7)+64260*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(
_Z^3+7)+49*_Z^2)*x^2-172200*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*x-4446*RootOf(_Z^3+7)*x^4
+17784*RootOf(_Z^3+7)*x^3+52326*RootOf(_Z^3+7)*x^2-140220*RootOf(_Z^3+7)*x+399*RootOf(36*RootOf(_Z^3+7)^2+42*_
Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)^3*x^4-1596*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*R
ootOf(_Z^3+7)^3*x^3-1638*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z
^2)*RootOf(_Z^3+7)*x+490*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)^2*RootOf(_Z^3+7)^2*x^4-1960*
RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)^2*RootOf(_Z^3+7)^2*x^3-980*RootOf(36*RootOf(_Z^3+7)^2
+42*_Z*RootOf(_Z^3+7)+49*_Z^2)^2*RootOf(_Z^3+7)^2*x^2-798*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_
Z^2)*RootOf(_Z^3+7)^3*x^2+5880*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)^2*RootOf(_Z^3+7)^2*x+4
788*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)^3*x+7560*(x^4-4*x^3-8*x^2+24*x+36)
^(2/3)*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)^2+7182*RootOf(_Z^3+7)^2*(x^4-4*
x^3-8*x^2+24*x+36)^(1/3)*x^2-14364*RootOf(_Z^3+7)^2*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)*x-4914*(x^4-4*x^3-8*x^2+24
*x+36)^(1/3)*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7))/(x^2-2*x-6)/(-1+x)^2)*Ro
otOf(_Z^3+7)-7/12*ln((-43092*RootOf(_Z^3+7)^2*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)+819*(x^4-4*x^3-8*x^2+24*x+36)^(1
/3)*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)*x^2+4914*(x^4-4*x^3-8*x^2+24*x+36)
^(2/3)-5460*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*x^4+21840*RootOf(36*RootOf(_Z^3+7)^2+42*_
Z*RootOf(_Z^3+7)+49*_Z^2)*x^3-320040*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)-260604*RootOf(_Z
^3+7)+64260*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*x^2-172200*RootOf(36*RootOf(_Z^3+7)^2+42*
_Z*RootOf(_Z^3+7)+49*_Z^2)*x-4446*RootOf(_Z^3+7)*x^4+17784*RootOf(_Z^3+7)*x^3+52326*RootOf(_Z^3+7)*x^2-140220*
RootOf(_Z^3+7)*x+399*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)^3*x^4-1596*RootOf
(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)^3*x^3-1638*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)*R
ootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)*x+490*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*
RootOf(_Z^3+7)+49*_Z^2)^2*RootOf(_Z^3+7)^2*x^4-1960*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)^2
*RootOf(_Z^3+7)^2*x^3-980*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)^2*RootOf(_Z^3+7)^2*x^2-798*
RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2)*RootOf(_Z^3+7)^3*x^2+5880*RootOf(36*RootOf(_Z^3+7)^2+
42*_Z*RootOf(_Z^3+7)+49*_Z^2)^2*RootOf(_Z^3+7)^2*x+4788*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^
2)*RootOf(_Z^3+7)^3*x+7560*(x^4-4*x^3-8*x^2+24*x+36)^(2/3)*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*
_Z^2)*RootOf(_Z^3+7)^2+7182*RootOf(_Z^3+7)^2*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)*x^2-14364*RootOf(_Z^3+7)^2*(x^4-4
*x^3-8*x^2+24*x+36)^(1/3)*x-4914*(x^4-4*x^3-8*x^2+24*x+36)^(1/3)*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+
7)+49*_Z^2)*RootOf(_Z^3+7))/(x^2-2*x-6)/(-1+x)^2)*RootOf(36*RootOf(_Z^3+7)^2+42*_Z*RootOf(_Z^3+7)+49*_Z^2))/(x
^2-2*x-6)^(2/3)*((x^2-2*x-6)^2)^(1/3)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} - 2 \, x - 6\right )}^{\frac {1}{3}}}{x - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-2*x-6)^(1/3)/(-1+x),x, algorithm="maxima")
[Out]
integrate((x^2 - 2*x - 6)^(1/3)/(x - 1), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^2-2\,x-6\right )}^{1/3}}{x-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2 - 2*x - 6)^(1/3)/(x - 1),x)
[Out]
int((x^2 - 2*x - 6)^(1/3)/(x - 1), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x^{2} - 2 x - 6}}{x - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**2-2*x-6)**(1/3)/(-1+x),x)
[Out]
Integral((x**2 - 2*x - 6)**(1/3)/(x - 1), x)
________________________________________________________________________________________