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

Optimal. Leaf size=175 \[ \frac {\sqrt {x^6-13 x^5+65 x^4-150 x^3+135 x^2+27 x-81} \left (8 x^2-62 x+115\right )}{24 (x-3)^2}-\frac {77}{16} \log \left (-2 x^3+13 x^2+2 \sqrt {x^6-13 x^5+65 x^4-150 x^3+135 x^2+27 x-81}-24 x+9\right )-8 \tan ^{-1}\left (\frac {x^2-6 x+9}{x^3-7 x^2-\sqrt {x^6-13 x^5+65 x^4-150 x^3+135 x^2+27 x-81}+15 x-9}\right )+\frac {77}{8} \log (x-3) \]

________________________________________________________________________________________

Rubi [A]  time = 0.33, antiderivative size = 208, normalized size of antiderivative = 1.19, number of steps used = 9, number of rules used = 9, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {6688, 6719, 1653, 814, 843, 621, 206, 724, 204} \begin {gather*} \frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} (41-18 x)}{8 (3-x)^2}-\frac {\left (-x^2+x+1\right ) \sqrt {-(3-x)^4 \left (-x^2+x+1\right )}}{3 (3-x)^2}+\frac {4 \sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )}{(3-x)^2 \sqrt {x^2-x-1}}-\frac {77 \sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )}{16 (3-x)^2 \sqrt {x^2-x-1}} \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]/(-1 + x),x]

[Out]

((41 - 18*x)*Sqrt[-((3 - x)^4*(1 + x - x^2))])/(8*(3 - x)^2) - ((1 + x - x^2)*Sqrt[-((3 - x)^4*(1 + x - x^2))]
)/(3*(3 - x)^2) + (4*Sqrt[-((3 - x)^4*(1 + x - x^2))]*ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])])/((3 - x)^2*Sqrt[
-1 - x + x^2]) - (77*Sqrt[-((3 - x)^4*(1 + x - x^2))]*ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])])/(16*(3 - x)^2
*Sqrt[-1 - x + x^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 {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx &=\int \frac {\sqrt {(-3+x)^4 \left (-1-x+x^2\right )}}{-1+x} \, dx\\ &=\frac {\sqrt {(-3+x)^4 \left (-1-x+x^2\right )} \int \frac {(-3+x)^2 \sqrt {-1-x+x^2}}{-1+x} \, dx}{(-3+x)^2 \sqrt {-1-x+x^2}}\\ &=-\frac {\left (1+x-x^2\right ) \sqrt {-(3-x)^4 \left (1+x-x^2\right )}}{3 (3-x)^2}+\frac {\sqrt {(-3+x)^4 \left (-1-x+x^2\right )} \int \frac {\left (\frac {51}{2}-\frac {27 x}{2}\right ) \sqrt {-1-x+x^2}}{-1+x} \, dx}{3 (-3+x)^2 \sqrt {-1-x+x^2}}\\ &=\frac {(41-18 x) \sqrt {-(3-x)^4 \left (1+x-x^2\right )}}{8 (3-x)^2}-\frac {\left (1+x-x^2\right ) \sqrt {-(3-x)^4 \left (1+x-x^2\right )}}{3 (3-x)^2}-\frac {\sqrt {(-3+x)^4 \left (-1-x+x^2\right )} \int \frac {\frac {423}{4}-\frac {231 x}{4}}{(-1+x) \sqrt {-1-x+x^2}} \, dx}{12 (-3+x)^2 \sqrt {-1-x+x^2}}\\ &=\frac {(41-18 x) \sqrt {-(3-x)^4 \left (1+x-x^2\right )}}{8 (3-x)^2}-\frac {\left (1+x-x^2\right ) \sqrt {-(3-x)^4 \left (1+x-x^2\right )}}{3 (3-x)^2}-\frac {\left (4 \sqrt {(-3+x)^4 \left (-1-x+x^2\right )}\right ) \int \frac {1}{(-1+x) \sqrt {-1-x+x^2}} \, dx}{(-3+x)^2 \sqrt {-1-x+x^2}}+\frac {\left (77 \sqrt {(-3+x)^4 \left (-1-x+x^2\right )}\right ) \int \frac {1}{\sqrt {-1-x+x^2}} \, dx}{16 (-3+x)^2 \sqrt {-1-x+x^2}}\\ &=\frac {(41-18 x) \sqrt {-(3-x)^4 \left (1+x-x^2\right )}}{8 (3-x)^2}-\frac {\left (1+x-x^2\right ) \sqrt {-(3-x)^4 \left (1+x-x^2\right )}}{3 (3-x)^2}+\frac {\left (8 \sqrt {(-3+x)^4 \left (-1-x+x^2\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3+x}{\sqrt {-1-x+x^2}}\right )}{(-3+x)^2 \sqrt {-1-x+x^2}}+\frac {\left (77 \sqrt {(-3+x)^4 \left (-1-x+x^2\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x}{\sqrt {-1-x+x^2}}\right )}{8 (-3+x)^2 \sqrt {-1-x+x^2}}\\ &=\frac {(41-18 x) \sqrt {-(3-x)^4 \left (1+x-x^2\right )}}{8 (3-x)^2}-\frac {\left (1+x-x^2\right ) \sqrt {-(3-x)^4 \left (1+x-x^2\right )}}{3 (3-x)^2}+\frac {4 \sqrt {-(3-x)^4 \left (1+x-x^2\right )} \tan ^{-1}\left (\frac {3-x}{2 \sqrt {-1-x+x^2}}\right )}{(3-x)^2 \sqrt {-1-x+x^2}}-\frac {77 \sqrt {-(3-x)^4 \left (1+x-x^2\right )} \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )}{16 (3-x)^2 \sqrt {-1-x+x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 161, normalized size = 0.92 \begin {gather*} \frac {(x-3)^2 \sqrt {x^2-x-1} \left (2 \left (8 \sqrt {x^2-x-1} x^2-62 \sqrt {x^2-x-1} x+115 \sqrt {x^2-x-1}+96 \tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )\right )-246 \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )-15 \tanh ^{-1}\left (\frac {2 x-1}{2 \sqrt {x^2-x-1}}\right )\right )}{48 \sqrt {(x-3)^4 \left (x^2-x-1\right )}} \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]/(-1 + x),x]

[Out]

((-3 + x)^2*Sqrt[-1 - x + x^2]*(2*(115*Sqrt[-1 - x + x^2] - 62*x*Sqrt[-1 - x + x^2] + 8*x^2*Sqrt[-1 - x + x^2]
+ 96*ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])]) - 246*ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])] - 15*ArcTanh[(-1
+ 2*x)/(2*Sqrt[-1 - x + x^2])]))/(48*Sqrt[(-3 + x)^4*(-1 - x + x^2)])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.45, size = 175, normalized size = 1.00 \begin {gather*} \frac {\left (115-62 x+8 x^2\right ) \sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{24 (-3+x)^2}-8 \tan ^{-1}\left (\frac {9-6 x+x^2}{-9+15 x-7 x^2+x^3-\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}\right )+\frac {77}{8} \log (-3+x)-\frac {77}{16} \log \left (9-24 x+13 x^2-2 x^3+2 \sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}\right ) \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]/(-1 + x),x]

[Out]

((115 - 62*x + 8*x^2)*Sqrt[-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6])/(24*(-3 + x)^2) - 8*ArcTan
[(9 - 6*x + x^2)/(-9 + 15*x - 7*x^2 + x^3 - Sqrt[-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6])] + (
77*Log[-3 + x])/8 - (77*Log[9 - 24*x + 13*x^2 - 2*x^3 + 2*Sqrt[-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^
5 + x^6]])/16

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 202, normalized size = 1.15 \begin {gather*} -\frac {205 \, x^{2} + 1536 \, {\left (x^{2} - 6 \, x + 9\right )} \arctan \left (-\frac {x^{3} - 7 \, x^{2} + 15 \, x - \sqrt {x^{6} - 13 \, x^{5} + 65 \, x^{4} - 150 \, x^{3} + 135 \, x^{2} + 27 \, x - 81} - 9}{x^{2} - 6 \, x + 9}\right ) + 924 \, {\left (x^{2} - 6 \, x + 9\right )} \log \left (-\frac {2 \, x^{3} - 13 \, x^{2} + 24 \, x - 2 \, \sqrt {x^{6} - 13 \, x^{5} + 65 \, x^{4} - 150 \, x^{3} + 135 \, x^{2} + 27 \, x - 81} - 9}{x^{2} - 6 \, x + 9}\right ) - 8 \, \sqrt {x^{6} - 13 \, x^{5} + 65 \, x^{4} - 150 \, x^{3} + 135 \, x^{2} + 27 \, x - 81} {\left (8 \, x^{2} - 62 \, x + 115\right )} - 1230 \, x + 1845}{192 \, {\left (x^{2} - 6 \, x + 9\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)^(1/2)/(-1+x),x, algorithm="fricas")

[Out]

-1/192*(205*x^2 + 1536*(x^2 - 6*x + 9)*arctan(-(x^3 - 7*x^2 + 15*x - sqrt(x^6 - 13*x^5 + 65*x^4 - 150*x^3 + 13
5*x^2 + 27*x - 81) - 9)/(x^2 - 6*x + 9)) + 924*(x^2 - 6*x + 9)*log(-(2*x^3 - 13*x^2 + 24*x - 2*sqrt(x^6 - 13*x
^5 + 65*x^4 - 150*x^3 + 135*x^2 + 27*x - 81) - 9)/(x^2 - 6*x + 9)) - 8*sqrt(x^6 - 13*x^5 + 65*x^4 - 150*x^3 +
135*x^2 + 27*x - 81)*(8*x^2 - 62*x + 115) - 1230*x + 1845)/(x^2 - 6*x + 9)

________________________________________________________________________________________

giac [A]  time = 0.38, size = 62, normalized size = 0.35 \begin {gather*} \frac {1}{24} \, {\left (2 \, {\left (4 \, x - 31\right )} x + 115\right )} \sqrt {x^{2} - x - 1} - 8 \, \arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) - \frac {77}{16} \, \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)^(1/2)/(-1+x),x, algorithm="giac")

[Out]

1/24*(2*(4*x - 31)*x + 115)*sqrt(x^2 - x - 1) - 8*arctan(-x + sqrt(x^2 - x - 1) + 1) - 77/16*log(abs(-2*x + 2*
sqrt(x^2 - x - 1) + 1))

________________________________________________________________________________________

maple [A]  time = 0.31, size = 102, normalized size = 0.58

method	result	size
risch	\(\frac {\left (8 x^{2}-62 x +115\right ) \sqrt {\left (x^{2}-x -1\right ) \left (-3+x \right )^{4}}}{24 \left (-3+x \right )^{2}}+\frac {\left (\frac {77 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x -1}\right )}{16}-4 \arctan \left (\frac {-3+x}{2 \sqrt {\left (-1+x \right )^{2}-2+x}}\right )\right ) \sqrt {\left (x^{2}-x -1\right ) \left (-3+x \right )^{4}}}{\left (-3+x \right )^{2} \sqrt {x^{2}-x -1}}\)	\(102\)
default	\(\frac {\sqrt {x^{6}-13 x^{5}+65 x^{4}-150 x^{3}+135 x^{2}+27 x -81}\, \left (16 \left (x^{2}-x -1\right )^{\frac {3}{2}}-108 x \sqrt {x^{2}-x -1}+246 \sqrt {x^{2}-x -1}+231 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x -1}\right )-192 \arctan \left (\frac {-3+x}{2 \sqrt {x^{2}-x -1}}\right )\right )}{48 \left (-3+x \right )^{2} \sqrt {x^{2}-x -1}}\)	\(120\)
trager	\(\frac {\left (8 x^{2}-62 x +115\right ) \sqrt {x^{6}-13 x^{5}+65 x^{4}-150 x^{3}+135 x^{2}+27 x -81}}{24 \left (-3+x \right )^{2}}+\frac {77 \ln \left (\frac {2 x^{3}-13 x^{2}+2 \sqrt {x^{6}-13 x^{5}+65 x^{4}-150 x^{3}+135 x^{2}+27 x -81}+24 x -9}{\left (-3+x \right )^{2}}\right )}{16}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-9 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+27 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -27 \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x^{6}-13 x^{5}+65 x^{4}-150 x^{3}+135 x^{2}+27 x -81}}{\left (-1+x \right ) \left (-3+x \right )^{2}}\right )\)	\(197\)

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

[Out]

1/24*(8*x^2-62*x+115)*((x^2-x-1)*(-3+x)^4)^(1/2)/(-3+x)^2+(77/16*ln(x-1/2+(x^2-x-1)^(1/2))-4*arctan(1/2*(-3+x)
/((-1+x)^2-2+x)^(1/2)))*((x^2-x-1)*(-3+x)^4)^(1/2)/(-3+x)^2/(x^2-x-1)^(1/2)

________________________________________________________________________________________

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x^6-13\,x^5+65\,x^4-150\,x^3+135\,x^2+27\,x-81}}{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)^(1/2)/(x - 1),x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x - 3\right )^{4} \left (x^{2} - x - 1\right )}}{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)**(1/2)/(-1+x),x)

[Out]

Integral(sqrt((x - 3)**4*(x**2 - x - 1))/(x - 1), x)

________________________________________________________________________________________