3.31.84 \(\int \frac {\sqrt {(-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6)^3}}{-1+x} \, dx\)

Optimal. Leaf size=518 \[ -\frac {19451047 \log \left (x^8-19 x^7+152 x^6-657 x^5+1620 x^4-2133 x^3+972 x^2+729 x-729\right )}{65536}+\frac {\sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^9+8764767 x^8+592677 x^7-10219851 x^6+9880866 x^5-885735 x^4-4704237 x^3+2480058 x^2+531441 x-531441} \left (1146880 x^8-23296000 x^7+199009280 x^6-910869760 x^5+2304529024 x^4-2700564848 x^3-508033624 x^2+4423205098 x-1245336401\right )}{10321920 (x-3)^6 \left (x^2-x-1\right )}+\frac {19451047 \log \left (-2 x^9+39 x^8-323 x^7+1466 x^6-3897 x^5+5886 x^4-4077 x^3-486 x^2+2187 x+2 \sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^9+8764767 x^8+592677 x^7-10219851 x^6+9880866 x^5-885735 x^4-4704237 x^3+2480058 x^2+531441 x-531441}-729\right )}{65536}+128 \tan ^{-1}\left (\frac {\left (x^2-x-1\right ) \left (x^6-18 x^5+135 x^4-540 x^3+1215 x^2-1458 x+729\right )}{x^9-20 x^8+171 x^7-809 x^6+2277 x^5-3753 x^4+3105 x^3-243 x^2-1458 x-\sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^9+8764767 x^8+592677 x^7-10219851 x^6+9880866 x^5-885735 x^4-4704237 x^3+2480058 x^2+531441 x-531441}+729}\right ) \]

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Rubi [A]  time = 0.93, antiderivative size = 459, normalized size of antiderivative = 0.89, number of steps used = 14, number of rules used = 9, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {6688, 6719, 1653, 814, 843, 621, 206, 724, 204} \begin {gather*} -\frac {\left (-x^2+x+1\right ) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} (1-x)^4}{9 (3-x)^6}-\frac {229 \left (-x^2+x+1\right ) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} (1-x)^3}{144 (3-x)^6}-\frac {19927 \left (-x^2+x+1\right ) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} (1-x)^2}{2016 (3-x)^6}-\frac {281233 \left (-x^2+x+1\right ) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} (1-x)}{8064 (3-x)^6}-\frac {6158183 \left (-x^2+x+1\right ) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3}}{80640 (3-x)^6}+\frac {(5567931-6941558 x) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3}}{32768 (3-x)^6 \left (-x^2+x+1\right )}+\frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3}}{12288 (3-x)^6}-\frac {64 \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )}{(3-x)^6 \left (x^2-x-1\right )^{3/2}}+\frac {19451047 \sqrt {-(3-x)^{12} \left (-x^2+x+1\right )^3} \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )}{65536 (3-x)^6 \left (x^2-x-1\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[(-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6)^3]/(-1 + x),x]

[Out]

((903871 - 1283454*x)*Sqrt[-((3 - x)^12*(1 + x - x^2)^3)])/(12288*(3 - x)^6) + ((5567931 - 6941558*x)*Sqrt[-((
3 - x)^12*(1 + x - x^2)^3)])/(32768*(3 - x)^6*(1 + x - x^2)) - (6158183*(1 + x - x^2)*Sqrt[-((3 - x)^12*(1 + x
- x^2)^3)])/(80640*(3 - x)^6) - (281233*(1 - x)*(1 + x - x^2)*Sqrt[-((3 - x)^12*(1 + x - x^2)^3)])/(8064*(3 -
x)^6) - (19927*(1 - x)^2*(1 + x - x^2)*Sqrt[-((3 - x)^12*(1 + x - x^2)^3)])/(2016*(3 - x)^6) - (229*(1 - x)^3
*(1 + x - x^2)*Sqrt[-((3 - x)^12*(1 + x - x^2)^3)])/(144*(3 - x)^6) - ((1 - x)^4*(1 + x - x^2)*Sqrt[-((3 - x)^
12*(1 + x - x^2)^3)])/(9*(3 - x)^6) - (64*Sqrt[-((3 - x)^12*(1 + x - x^2)^3)]*ArcTan[(3 - x)/(2*Sqrt[-1 - x +
x^2])])/((3 - x)^6*(-1 - x + x^2)^(3/2)) + (19451047*Sqrt[-((3 - x)^12*(1 + x - x^2)^3)]*ArcTanh[(1 - 2*x)/(2*
Sqrt[-1 - x + x^2])])/(65536*(3 - x)^6*(-1 - x + x^2)^(3/2))

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 206

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

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {\left (-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right )^3}}{-1+x} \, dx &=\int \frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}}{-1+x} \, dx\\ &=\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {(-3+x)^6 \left (-1-x+x^2\right )^{3/2}}{-1+x} \, dx}{(-3+x)^6 \left (-1-x+x^2\right )^{3/2}}\\ &=-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (-1-x+x^2\right )^{3/2} \left (\frac {13125}{2}-\frac {26233 x}{2}+10889 x^2-4761 x^3+\frac {2233 x^4}{2}-\frac {229 x^5}{2}\right )}{-1+x} \, dx}{9 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}}\\ &=-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (-1-x+x^2\right )^{3/2} \left (\frac {210229}{4}-\frac {207803 x}{2}+82761 x^2-\frac {63581 x^3}{2}+\frac {19927 x^4}{4}\right )}{-1+x} \, dx}{72 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}}\\ &=-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (-1-x+x^2\right )^{3/2} \left (\frac {2923279}{8}-\frac {5400017 x}{8}+\frac {3578485 x^2}{8}-\frac {843699 x^3}{8}\right )}{-1+x} \, dx}{504 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}}\\ &=-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (-1-x+x^2\right )^{3/2} \left (\frac {32548251}{16}-2995389 x+\frac {18474549 x^2}{16}\right )}{-1+x} \, dx}{3024 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}}\\ &=-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (\frac {233109765}{32}-\frac {202144005 x}{32}\right ) \left (-1-x+x^2\right )^{3/2}}{-1+x} \, dx}{15120 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}}\\ &=\frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{12288 (3-x)^6}-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}-\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\left (\frac {3775338315}{64}-\frac {3279886155 x}{64}\right ) \sqrt {-1-x+x^2}}{-1+x} \, dx}{120960 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}}\\ &=\frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{12288 (3-x)^6}+\frac {(5567931-6941558 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{32768 (3-x)^6 \left (1+x-x^2\right )}-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3} \int \frac {\frac {22344856695}{128}-\frac {18381239415 x}{128}}{(-1+x) \sqrt {-1-x+x^2}} \, dx}{483840 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}}\\ &=\frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{12288 (3-x)^6}+\frac {(5567931-6941558 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{32768 (3-x)^6 \left (1+x-x^2\right )}-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}+\frac {\left (64 \sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}\right ) \int \frac {1}{(-1+x) \sqrt {-1-x+x^2}} \, dx}{(-3+x)^6 \left (-1-x+x^2\right )^{3/2}}-\frac {\left (19451047 \sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}\right ) \int \frac {1}{\sqrt {-1-x+x^2}} \, dx}{65536 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}}\\ &=\frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{12288 (3-x)^6}+\frac {(5567931-6941558 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{32768 (3-x)^6 \left (1+x-x^2\right )}-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}-\frac {\left (128 \sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3+x}{\sqrt {-1-x+x^2}}\right )}{(-3+x)^6 \left (-1-x+x^2\right )^{3/2}}-\frac {\left (19451047 \sqrt {(-3+x)^{12} \left (-1-x+x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x}{\sqrt {-1-x+x^2}}\right )}{32768 (-3+x)^6 \left (-1-x+x^2\right )^{3/2}}\\ &=\frac {(903871-1283454 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{12288 (3-x)^6}+\frac {(5567931-6941558 x) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{32768 (3-x)^6 \left (1+x-x^2\right )}-\frac {6158183 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{80640 (3-x)^6}-\frac {281233 (1-x) \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{8064 (3-x)^6}-\frac {19927 (1-x)^2 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{2016 (3-x)^6}-\frac {229 (1-x)^3 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{144 (3-x)^6}-\frac {(1-x)^4 \left (1+x-x^2\right ) \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3}}{9 (3-x)^6}-\frac {64 \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3} \tan ^{-1}\left (\frac {3-x}{2 \sqrt {-1-x+x^2}}\right )}{(3-x)^6 \left (-1-x+x^2\right )^{3/2}}+\frac {19451047 \sqrt {-(3-x)^{12} \left (1+x-x^2\right )^3} \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )}{65536 (3-x)^6 \left (-1-x+x^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.40, size = 144, normalized size = 0.28 \begin {gather*} \frac {(x-3)^6 \left (x^2-x-1\right )^{3/2} \left (-1321205760 \tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )[phone] \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )+2 \sqrt {x^2-x-1} \left (1146880 x^8-23296000 x^7+199009280 x^6-910869760 x^5+2304529024 x^4-2700564848 x^3-508033624 x^2+4423205098 x-1245336401\right )\right )}{20643840 \sqrt {(x-3)^{12} \left (x^2-x-1\right )^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6)^3]/(-1 + x),x]

[Out]

((-3 + x)^6*(-1 - x + x^2)^(3/2)*(2*Sqrt[-1 - x + x^2]*(-1245336401 + 4423205098*x - 508033624*x^2 - 270056484
8*x^3 + 2304529024*x^4 - 910869760*x^5 + 199009280*x^6 - 23296000*x^7 + 1146880*x^8) - 1321205760*ArcTan[(3 -
x)/(2*Sqrt[-1 - x + x^2])] + 6127079805*ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])]))/(20643840*Sqrt[(-3 + x)^12
*(-1 - x + x^2)^3])

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IntegrateAlgebraic [A]  time = 0.43, size = 518, normalized size = 1.00 \begin {gather*} \frac {\left (-1245336401+4423205098 x-508033624 x^2-2700564848 x^3+2304529024 x^4-910869760 x^5+199009280 x^6-23296000 x^7+1146880 x^8\right ) \sqrt {-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}}{10321920 (-3+x)^6 \left (-1-x+x^2\right )}+128 \tan ^{-1}\left (\frac {\left (-1-x+x^2\right ) \left (729-1458 x+1215 x^2-540 x^3+135 x^4-18 x^5+x^6\right )}{729-1458 x-243 x^2+3105 x^3-3753 x^4+2277 x^5-809 x^6+171 x^7-20 x^8+x^9-\sqrt {-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}}\right )-\frac {19451047 \log \left (-729+729 x+972 x^2-2133 x^3+1620 x^4-657 x^5+152 x^6-19 x^7+x^8\right )}{65536}+\frac {19451047 \log \left (-729+2187 x-486 x^2-4077 x^3+5886 x^4-3897 x^5+1466 x^6-323 x^7+39 x^8-2 x^9+2 \sqrt {-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}\right )}{65536} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[(-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6)^3]/(-1 + x),x]

[Out]

((-1245336401 + 4423205098*x - 508033624*x^2 - 2700564848*x^3 + 2304529024*x^4 - 910869760*x^5 + 199009280*x^6
- 23296000*x^7 + 1146880*x^8)*Sqrt[-531441 + 531441*x + 2480058*x^2 - 4704237*x^3 - 885735*x^4 + 9880866*x^5
- 10219851*x^6 + 592677*x^7 + 8764767*x^8 - 10819710*x^9 + 7498953*x^10 - 3554163*x^11 + 1221371*x^12 - 309774
*x^13 + 57735*x^14 - 7717*x^15 + 702*x^16 - 39*x^17 + x^18])/(10321920*(-3 + x)^6*(-1 - x + x^2)) + 128*ArcTan
[((-1 - x + x^2)*(729 - 1458*x + 1215*x^2 - 540*x^3 + 135*x^4 - 18*x^5 + x^6))/(729 - 1458*x - 243*x^2 + 3105*
x^3 - 3753*x^4 + 2277*x^5 - 809*x^6 + 171*x^7 - 20*x^8 + x^9 - Sqrt[-531441 + 531441*x + 2480058*x^2 - 4704237
*x^3 - 885735*x^4 + 9880866*x^5 - 10219851*x^6 + 592677*x^7 + 8764767*x^8 - 10819710*x^9 + 7498953*x^10 - 3554
163*x^11 + 1221371*x^12 - 309774*x^13 + 57735*x^14 - 7717*x^15 + 702*x^16 - 39*x^17 + x^18])] - (19451047*Log[
-729 + 729*x + 972*x^2 - 2133*x^3 + 1620*x^4 - 657*x^5 + 152*x^6 - 19*x^7 + x^8])/65536 + (19451047*Log[-729 +
2187*x - 486*x^2 - 4077*x^3 + 5886*x^4 - 3897*x^5 + 1466*x^6 - 323*x^7 + 39*x^8 - 2*x^9 + 2*Sqrt[-531441 + 53
1441*x + 2480058*x^2 - 4704237*x^3 - 885735*x^4 + 9880866*x^5 - 10219851*x^6 + 592677*x^7 + 8764767*x^8 - 1081
9710*x^9 + 7498953*x^10 - 3554163*x^11 + 1221371*x^12 - 309774*x^13 + 57735*x^14 - 7717*x^15 + 702*x^16 - 39*x
^17 + x^18]])/65536

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fricas [A]  time = 0.51, size = 652, normalized size = 1.26 \begin {gather*} \frac {4819349233 \, x^{8} - 91567635427 \, x^{7} + 732541083416 \, x^{6} - 3166312446081 \, x^{5} + 7807345757460 \, x^{4} - 10279671913989 \, x^{3} + 4684407454476 \, x^{2} + 42278584320 \, {\left (x^{8} - 19 \, x^{7} + 152 \, x^{6} - 657 \, x^{5} + 1620 \, x^{4} - 2133 \, x^{3} + 972 \, x^{2} + 729 \, x - 729\right )} \arctan \left (-\frac {x^{9} - 20 \, x^{8} + 171 \, x^{7} - 809 \, x^{6} + 2277 \, x^{5} - 3753 \, x^{4} + 3105 \, x^{3} - 243 \, x^{2} - 1458 \, x - \sqrt {x^{18} - 39 \, x^{17} + 702 \, x^{16} - 7717 \, x^{15} + 57735 \, x^{14} - 309774 \, x^{13} + 1221371 \, x^{12} - 3554163 \, x^{11} + 7498953 \, x^{10} - 10819710 \, x^{9} + 8764767 \, x^{8} + 592677 \, x^{7} - 10219851 \, x^{6} + 9880866 \, x^{5} - 885735 \, x^{4} - 4704237 \, x^{3} + 2480058 \, x^{2} + 531441 \, x - 531441} + 729}{x^{8} - 19 \, x^{7} + 152 \, x^{6} - 657 \, x^{5} + 1620 \, x^{4} - 2133 \, x^{3} + 972 \, x^{2} + 729 \, x - 729}\right ) + 98033276880 \, {\left (x^{8} - 19 \, x^{7} + 152 \, x^{6} - 657 \, x^{5} + 1620 \, x^{4} - 2133 \, x^{3} + 972 \, x^{2} + 729 \, x - 729\right )} \log \left (-\frac {2 \, x^{9} - 39 \, x^{8} + 323 \, x^{7} - 1466 \, x^{6} + 3897 \, x^{5} - 5886 \, x^{4} + 4077 \, x^{3} + 486 \, x^{2} - 2187 \, x - 2 \, \sqrt {x^{18} - 39 \, x^{17} + 702 \, x^{16} - 7717 \, x^{15} + 57735 \, x^{14} - 309774 \, x^{13} + 1221371 \, x^{12} - 3554163 \, x^{11} + 7498953 \, x^{10} - 10819710 \, x^{9} + 8764767 \, x^{8} + 592677 \, x^{7} - 10219851 \, x^{6} + 9880866 \, x^{5} - 885735 \, x^{4} - 4704237 \, x^{3} + 2480058 \, x^{2} + 531441 \, x - 531441} + 729}{x^{8} - 19 \, x^{7} + 152 \, x^{6} - 657 \, x^{5} + 1620 \, x^{4} - 2133 \, x^{3} + 972 \, x^{2} + 729 \, x - 729}\right ) + 32 \, \sqrt {x^{18} - 39 \, x^{17} + 702 \, x^{16} - 7717 \, x^{15} + 57735 \, x^{14} - 309774 \, x^{13} + 1221371 \, x^{12} - 3554163 \, x^{11} + 7498953 \, x^{10} - 10819710 \, x^{9} + 8764767 \, x^{8} + 592677 \, x^{7} - 10219851 \, x^{6} + 9880866 \, x^{5} - 885735 \, x^{4} - 4704237 \, x^{3} + 2480058 \, x^{2} + 531441 \, x - 531441} {\left (1146880 \, x^{8} - 23296000 \, x^{7} + 199009280 \, x^{6} - 910869760 \, x^{5} + 2304529024 \, x^{4} - 2700564848 \, x^{3} - 508033624 \, x^{2} + 4423205098 \, x - 1245336401\right )} + 3513305590857 \, x - 3513305590857}{330301440 \, {\left (x^{8} - 19 \, x^{7} + 152 \, x^{6} - 657 \, x^{5} + 1620 \, x^{4} - 2133 \, x^{3} + 972 \, x^{2} + 729 \, x - 729\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6-13*x^5+65*x^4-150*x^3+135*x^2+27*x-81)^3)^(1/2)/(-1+x),x, algorithm="fricas")

[Out]

1/330301440*(4819349233*x^8 - 91567635427*x^7 + 732541083416*x^6 - 3166312446081*x^5 + 7807345757460*x^4 - 102
79671913989*x^3 + 4684407454476*x^2 + 42278584320*(x^8 - 19*x^7 + 152*x^6 - 657*x^5 + 1620*x^4 - 2133*x^3 + 97
2*x^2 + 729*x - 729)*arctan(-(x^9 - 20*x^8 + 171*x^7 - 809*x^6 + 2277*x^5 - 3753*x^4 + 3105*x^3 - 243*x^2 - 14
58*x - sqrt(x^18 - 39*x^17 + 702*x^16 - 7717*x^15 + 57735*x^14 - 309774*x^13 + 1221371*x^12 - 3554163*x^11 + 7
498953*x^10 - 10819710*x^9 + 8764767*x^8 + 592677*x^7 - 10219851*x^6 + 9880866*x^5 - 885735*x^4 - 4704237*x^3
+ 2480058*x^2 + 531441*x - 531441) + 729)/(x^8 - 19*x^7 + 152*x^6 - 657*x^5 + 1620*x^4 - 2133*x^3 + 972*x^2 +
729*x - 729)) + 98033276880*(x^8 - 19*x^7 + 152*x^6 - 657*x^5 + 1620*x^4 - 2133*x^3 + 972*x^2 + 729*x - 729)*l
og(-(2*x^9 - 39*x^8 + 323*x^7 - 1466*x^6 + 3897*x^5 - 5886*x^4 + 4077*x^3 + 486*x^2 - 2187*x - 2*sqrt(x^18 - 3
9*x^17 + 702*x^16 - 7717*x^15 + 57735*x^14 - 309774*x^13 + 1221371*x^12 - 3554163*x^11 + 7498953*x^10 - 108197
10*x^9 + 8764767*x^8 + 592677*x^7 - 10219851*x^6 + 9880866*x^5 - 885735*x^4 - 4704237*x^3 + 2480058*x^2 + 5314
41*x - 531441) + 729)/(x^8 - 19*x^7 + 152*x^6 - 657*x^5 + 1620*x^4 - 2133*x^3 + 972*x^2 + 729*x - 729)) + 32*s
qrt(x^18 - 39*x^17 + 702*x^16 - 7717*x^15 + 57735*x^14 - 309774*x^13 + 1221371*x^12 - 3554163*x^11 + 7498953*x
^10 - 10819710*x^9 + 8764767*x^8 + 592677*x^7 - 10219851*x^6 + 9880866*x^5 - 885735*x^4 - 4704237*x^3 + 248005
8*x^2 + 531441*x - 531441)*(1146880*x^8 - 23296000*x^7 + 199009280*x^6 - 910869760*x^5 + 2304529024*x^4 - 2700
564848*x^3 - 508033624*x^2 + 4423205098*x - 1245336401) + 3513305590857*x - 3513305590857)/(x^8 - 19*x^7 + 152
*x^6 - 657*x^5 + 1620*x^4 - 2133*x^3 + 972*x^2 + 729*x - 729)

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giac [A]  time = 0.33, size = 92, normalized size = 0.18 \begin {gather*} \frac {1}{10321920} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, {\left (16 \, x - 325\right )} x + 38869\right )} x - 711617\right )} x + 18004133\right )} x - 168785303\right )} x - 63504203\right )} x + 2211602549\right )} x - 1245336401\right )} \sqrt {x^{2} - x - 1} + 128 \, \arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) + \frac {19451047}{65536} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6-13*x^5+65*x^4-150*x^3+135*x^2+27*x-81)^3)^(1/2)/(-1+x),x, algorithm="giac")

[Out]

1/10321920*(2*(4*(2*(8*(10*(4*(14*(16*x - 325)*x + 38869)*x - 711617)*x + 18004133)*x - 168785303)*x - 6350420
3)*x + 2211602549)*x - 1245336401)*sqrt(x^2 - x - 1) + 128*arctan(-x + sqrt(x^2 - x - 1) + 1) + 19451047/65536
*log(abs(-2*x + 2*sqrt(x^2 - x - 1) + 1))

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maple [A]  time = 0.44, size = 146, normalized size = 0.28

method	result	size
risch	\(\frac {\left (1146880 x^{8}-23296000 x^{7}[phone] x^{6}-910869760 x^{5}[phone] x^{4}-2700564848 x^{3}-508033624 x^{2}[phone] x -1245336401\right ) \sqrt {\left (x^{2}-x -1\right )^{3} \left (-3+x \right )^{12}}}{10321920 \left (x^{2}-x -1\right ) \left (-3+x \right )^{6}}+\frac {\left (-\frac {19451047 \ln \left (-\frac {1}{2}+x +\sqrt {x^{2}-x -1}\right )}{65536}+64 \arctan \left (\frac {-3+x}{2 \sqrt {\left (-1+x \right )^{2}-2+x}}\right )\right ) \sqrt {\left (x^{2}-x -1\right )^{3} \left (-3+x \right )^{12}}}{\left (x^{2}-x -1\right )^{\frac {3}{2}} \left (-3+x \right )^{6}}\)	\(146\)
default	\(\frac {\sqrt {\left (x^{6}-13 x^{5}+65 x^{4}-150 x^{3}+135 x^{2}+27 x -81\right )^{3}}\, \left (2293760 x^{4} \left (x^{2}-x -1\right )^{\frac {5}{2}}-42004480 x^{3} \left (x^{2}-x -1\right )^{\frac {5}{2}}[phone] x^{2} \left (x^{2}-x -1\right )^{\frac {5}{2}}-1235724800 x \left (x^{2}-x -1\right )^{\frac {5}{2}}[phone] \left (x^{2}-x -1\right )^{\frac {5}{2}}-2156202720 x \left (x^{2}-x -1\right )^{\frac {3}{2}}[phone] \left (x^{2}-x -1\right )^{\frac {3}{2}}[phone] x \sqrt {x^{2}-x -1}-3507796530 \sqrt {x^{2}-x -1}-6127079805 \ln \left (-\frac {1}{2}+x +\sqrt {x^{2}-x -1}\right )[phone] \arctan \left (\frac {-3+x}{2 \sqrt {x^{2}-x -1}}\right )\right )}{20643840 \left (-3+x \right )^{6} \left (x^{2}-x -1\right )^{\frac {3}{2}}}\)	\(205\)
trager	\(\frac {\left (1146880 x^{8}-23296000 x^{7}[phone] x^{6}-910869760 x^{5}[phone] x^{4}-2700564848 x^{3}-508033624 x^{2}[phone] x -1245336401\right ) \sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^{9}+8764767 x^{8}+592677 x^{7}-10219851 x^{6}+9880866 x^{5}-885735 x^{4}-4704237 x^{3}+2480058 x^{2}+531441 x -531441}}{10321920 \left (-3+x \right )^{6} \left (x^{2}-x -1\right )}+\frac {19451047 \ln \left (-\frac {-729+2187 x -486 x^{2}-4077 x^{3}+5886 x^{4}-3897 x^{5}+1466 x^{6}-323 x^{7}+39 x^{8}-2 x^{9}+2 \sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^{9}+8764767 x^{8}+592677 x^{7}-10219851 x^{6}+9880866 x^{5}-885735 x^{4}-4704237 x^{3}+2480058 x^{2}+531441 x -531441}}{\left (x^{2}-x -1\right ) \left (-3+x \right )^{6}}\right )}{65536}+64 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{9}+22 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{8}-209 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{7}+1113 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-3591 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}+6993 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-7371 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2187 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2916 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {x^{18}-39 x^{17}+702 x^{16}-7717 x^{15}+57735 x^{14}-309774 x^{13}+1221371 x^{12}-3554163 x^{11}+7498953 x^{10}-10819710 x^{9}+8764767 x^{8}+592677 x^{7}-10219851 x^{6}+9880866 x^{5}-885735 x^{4}-4704237 x^{3}+2480058 x^{2}+531441 x -531441}-2187 \RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (x^{2}-x -1\right ) \left (-3+x \right )^{6}}\right )\)	\(534\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6-13*x^5+65*x^4-150*x^3+135*x^2+27*x-81)^3)^(1/2)/(-1+x),x,method=_RETURNVERBOSE)

[Out]

1/10321920/(x^2-x-1)*(1146880*x^8-23296000*x^7+199009280*x^6-910869760*x^5+2304529024*x^4-2700564848*x^3-50803
3624*x^2+4423205098*x-1245336401)*((x^2-x-1)^3*(-3+x)^12)^(1/2)/(-3+x)^6+(-19451047/65536*ln(-1/2+x+(x^2-x-1)^
(1/2))+64*arctan(1/2*(-3+x)/((-1+x)^2-2+x)^(1/2)))*((x^2-x-1)^3*(-3+x)^12)^(1/2)/(x^2-x-1)^(3/2)/(-3+x)^6

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {{\left (x^{6} - 13 \, x^{5} + 65 \, x^{4} - 150 \, x^{3} + 135 \, x^{2} + 27 \, x - 81\right )}^{3}}}{x - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^6-13*x^5+65*x^4-150*x^3+135*x^2+27*x-81)^3)^(1/2)/(-1+x),x, algorithm="maxima")

[Out]

integrate(sqrt((x^6 - 13*x^5 + 65*x^4 - 150*x^3 + 135*x^2 + 27*x - 81)^3)/(x - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {{\left (x^6-13\,x^5+65\,x^4-150\,x^3+135\,x^2+27\,x-81\right )}^3}}{x-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6 - 81)^3)^(1/2)/(x - 1),x)

[Out]

int(((27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6 - 81)^3)^(1/2)/(x - 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x - 3\right )^{12} \left (x^{2} - x - 1\right )^{3}}}{x - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**6-13*x**5+65*x**4-150*x**3+135*x**2+27*x-81)**3)**(1/2)/(-1+x),x)

[Out]

Integral(sqrt((x - 3)**12*(x**2 - x - 1)**3)/(x - 1), x)

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