\(\int \frac {\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx\) [2295]

Optimal result

Integrand size = 38, antiderivative size = 175 \[ \int \frac {\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx=\frac {\left (115-62 x+8 x^2\right ) \sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{24 (-3+x)^2}-8 \arctan \left (\frac {9-6 x+x^2}{-9+15 x-7 x^2+x^3-\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}\right )+\frac {77}{8} \log (-3+x)-\frac {77}{16} \log \left (9-24 x+13 x^2-2 x^3+2 \sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}\right ) \] Output:

1/24*(8*x^2-62*x+115)*(x^6-13*x^5+65*x^4-150*x^3+135*x^2+27*x-81)^(1/2)/(- 
3+x)^2-8*arctan((x^2-6*x+9)/(-9+15*x-7*x^2+x^3-(x^6-13*x^5+65*x^4-150*x^3+ 
135*x^2+27*x-81)^(1/2)))+77/8*ln(-3+x)-77/16*ln(9-24*x+13*x^2-2*x^3+2*(x^6 
-13*x^5+65*x^4-150*x^3+135*x^2+27*x-81)^(1/2))
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx=\frac {(-3+x)^2 \sqrt {-1-x+x^2} \left (2 \sqrt {-1-x+x^2} \left (115-62 x+8 x^2\right )-384 \arctan \left (1-x+\sqrt {-1-x+x^2}\right )-231 \log \left (1-2 x+2 \sqrt {-1-x+x^2}\right )\right )}{48 \sqrt {(-3+x)^4 \left (-1-x+x^2\right )}} \] Input:

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

Output:

((-3 + x)^2*Sqrt[-1 - x + x^2]*(2*Sqrt[-1 - x + x^2]*(115 - 62*x + 8*x^2) 
- 384*ArcTan[1 - x + Sqrt[-1 - x + x^2]] - 231*Log[1 - 2*x + 2*Sqrt[-1 - x 
 + x^2]]))/(48*Sqrt[(-3 + x)^4*(-1 - x + x^2)])
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.79, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {7239, 7270, 25, 1267, 27, 1231, 27, 1269, 1092, 219, 1154, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt {(x-3)^4 \left (x^2-x-1\right )}}{x-1}dx\)

\(\Big \downarrow \) 7270

\(\displaystyle \frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \int -\frac {(3-x)^2 \sqrt {x^2-x-1}}{1-x}dx}{(3-x)^2 \sqrt {x^2-x-1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \int \frac {(3-x)^2 \sqrt {x^2-x-1}}{1-x}dx}{(3-x)^2 \sqrt {x^2-x-1}}\)

\(\Big \downarrow \) 1267

\(\displaystyle -\frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \left (\frac {1}{3} \int \frac {3 (17-9 x) \sqrt {x^2-x-1}}{2 (1-x)}dx-\frac {1}{3} \left (x^2-x-1\right )^{3/2}\right )}{(3-x)^2 \sqrt {x^2-x-1}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \left (\frac {1}{2} \int \frac {(17-9 x) \sqrt {x^2-x-1}}{1-x}dx-\frac {1}{3} \left (x^2-x-1\right )^{3/2}\right )}{(3-x)^2 \sqrt {x^2-x-1}}\)

\(\Big \downarrow \) 1231

\(\displaystyle -\frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \left (\frac {1}{2} \left (-\frac {1}{4} \int \frac {141-77 x}{2 (1-x) \sqrt {x^2-x-1}}dx-\frac {1}{4} \sqrt {x^2-x-1} (41-18 x)\right )-\frac {1}{3} \left (x^2-x-1\right )^{3/2}\right )}{(3-x)^2 \sqrt {x^2-x-1}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \left (\frac {1}{2} \left (-\frac {1}{8} \int \frac {141-77 x}{(1-x) \sqrt {x^2-x-1}}dx-\frac {1}{4} \sqrt {x^2-x-1} (41-18 x)\right )-\frac {1}{3} \left (x^2-x-1\right )^{3/2}\right )}{(3-x)^2 \sqrt {x^2-x-1}}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \left (\frac {1}{2} \left (\frac {1}{8} \left (-77 \int \frac {1}{\sqrt {x^2-x-1}}dx-64 \int \frac {1}{(1-x) \sqrt {x^2-x-1}}dx\right )-\frac {1}{4} (41-18 x) \sqrt {x^2-x-1}\right )-\frac {1}{3} \left (x^2-x-1\right )^{3/2}\right )}{(3-x)^2 \sqrt {x^2-x-1}}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \left (\frac {1}{2} \left (\frac {1}{8} \left (-64 \int \frac {1}{(1-x) \sqrt {x^2-x-1}}dx-154 \int \frac {1}{4-\frac {(1-2 x)^2}{x^2-x-1}}d\left (-\frac {1-2 x}{\sqrt {x^2-x-1}}\right )\right )-\frac {1}{4} (41-18 x) \sqrt {x^2-x-1}\right )-\frac {1}{3} \left (x^2-x-1\right )^{3/2}\right )}{(3-x)^2 \sqrt {x^2-x-1}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \left (\frac {1}{2} \left (\frac {1}{8} \left (77 \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )-64 \int \frac {1}{(1-x) \sqrt {x^2-x-1}}dx\right )-\frac {1}{4} (41-18 x) \sqrt {x^2-x-1}\right )-\frac {1}{3} \left (x^2-x-1\right )^{3/2}\right )}{(3-x)^2 \sqrt {x^2-x-1}}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \left (\frac {1}{2} \left (\frac {1}{8} \left (128 \int \frac {1}{-\frac {(3-x)^2}{x^2-x-1}-4}d\frac {3-x}{\sqrt {x^2-x-1}}+77 \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )\right )-\frac {1}{4} (41-18 x) \sqrt {x^2-x-1}\right )-\frac {1}{3} \left (x^2-x-1\right )^{3/2}\right )}{(3-x)^2 \sqrt {x^2-x-1}}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\sqrt {-(3-x)^4 \left (-x^2+x+1\right )} \left (\frac {1}{2} \left (\frac {1}{8} \left (77 \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )-64 \arctan \left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )\right )-\frac {1}{4} (41-18 x) \sqrt {x^2-x-1}\right )-\frac {1}{3} \left (x^2-x-1\right )^{3/2}\right )}{(3-x)^2 \sqrt {x^2-x-1}}\)

Input:

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

Output:

-((Sqrt[-((3 - x)^4*(1 + x - x^2))]*(-1/3*(-1 - x + x^2)^(3/2) + (-1/4*((4 
1 - 18*x)*Sqrt[-1 - x + x^2]) + (-64*ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2]) 
] + 77*ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])])/8)/2))/((3 - x)^2*Sqrt[- 
1 - x + x^2]))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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

rule 219

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

rule 1092

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

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-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]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(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] - Simp[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] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1267

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

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(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] &&  !IGtQ[m, 0]
 

rule 7239

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

rule 7270

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.58

method	result	size
risch	\(\frac {\left (8 x^{2}-62 x +115\right ) \sqrt {\left (x^{2}-x -1\right ) \left (-3+x \right )^{4}}}{24 \left (-3+x \right )^{2}}+\frac {\left (\frac {77 \ln \left (-\frac {1}{2}+x +\sqrt {x^{2}-x -1}\right )}{16}-4 \arctan \left (\frac {-3+x}{2 \sqrt {\left (-1+x \right )^{2}-2+x}}\right )\right ) \sqrt {\left (x^{2}-x -1\right ) \left (-3+x \right )^{4}}}{\left (-3+x \right )^{2} \sqrt {x^{2}-x -1}}\)	\(102\)
default	\(\frac {\sqrt {x^{6}-13 x^{5}+65 x^{4}-150 x^{3}+135 x^{2}+27 x -81}\, \left (16 \left (x^{2}-x -1\right )^{\frac {3}{2}}-108 x \sqrt {x^{2}-x -1}+246 \sqrt {x^{2}-x -1}+231 \ln \left (-\frac {1}{2}+x +\sqrt {x^{2}-x -1}\right )-192 \arctan \left (\frac {-3+x}{2 \sqrt {x^{2}-x -1}}\right )\right )}{48 \left (-3+x \right )^{2} \sqrt {x^{2}-x -1}}\)	\(120\)
trager	\(\frac {\left (8 x^{2}-62 x +115\right ) \sqrt {x^{6}-13 x^{5}+65 x^{4}-150 x^{3}+135 x^{2}+27 x -81}}{24 \left (-3+x \right )^{2}}-\frac {77 \ln \left (\frac {9-24 x +13 x^{2}-2 x^{3}+2 \sqrt {x^{6}-13 x^{5}+65 x^{4}-150 x^{3}+135 x^{2}+27 x -81}}{\left (-3+x \right )^{2}}\right )}{16}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -27 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \sqrt {x^{6}-13 x^{5}+65 x^{4}-150 x^{3}+135 x^{2}+27 x -81}}{\left (-1+x \right ) \left (-3+x \right )^{2}}\right )\)	\(196\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx=-\frac {205 \, x^{2} + 1536 \, {\left (x^{2} - 6 \, x + 9\right )} \arctan \left (-\frac {x^{3} - 7 \, x^{2} + 15 \, x - \sqrt {x^{6} - 13 \, x^{5} + 65 \, x^{4} - 150 \, x^{3} + 135 \, x^{2} + 27 \, x - 81} - 9}{x^{2} - 6 \, x + 9}\right ) + 924 \, {\left (x^{2} - 6 \, x + 9\right )} \log \left (-\frac {2 \, x^{3} - 13 \, x^{2} + 24 \, x - 2 \, \sqrt {x^{6} - 13 \, x^{5} + 65 \, x^{4} - 150 \, x^{3} + 135 \, x^{2} + 27 \, x - 81} - 9}{x^{2} - 6 \, x + 9}\right ) - 8 \, \sqrt {x^{6} - 13 \, x^{5} + 65 \, x^{4} - 150 \, x^{3} + 135 \, x^{2} + 27 \, x - 81} {\left (8 \, x^{2} - 62 \, x + 115\right )} - 1230 \, x + 1845}{192 \, {\left (x^{2} - 6 \, x + 9\right )}} \] Input:

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

Output:

-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 + 135*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)
 

Sympy [F]

\[ \int \frac {\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx=\int \frac {\sqrt {\left (x - 3\right )^{4} \left (x^{2} - x - 1\right )}}{x - 1}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx=\int { \frac {\sqrt {x^{6} - 13 \, x^{5} + 65 \, x^{4} - 150 \, x^{3} + 135 \, x^{2} + 27 \, x - 81}}{x - 1} \,d x } \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx=\frac {1}{24} \, {\left (2 \, {\left (4 \, x - 31\right )} x + 115\right )} \sqrt {x^{2} - x - 1} - 8 \, \arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) - \frac {77}{16} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \] Input:

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

Output:

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))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx=\int \frac {\sqrt {x^6-13\,x^5+65\,x^4-150\,x^3+135\,x^2+27\,x-81}}{x-1} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx=-8 \mathit {atan} \left (\sqrt {x^{2}-x -1}+x -1\right )+\frac {\sqrt {x^{2}-x -1}\, x^{2}}{3}-\frac {31 \sqrt {x^{2}-x -1}\, x}{12}+\frac {115 \sqrt {x^{2}-x -1}}{24}+\frac {77 \,\mathrm {log}\left (\frac {2 \sqrt {x^{2}-x -1}+2 x -1}{\sqrt {5}}\right )}{16} \] Input:

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

Output:

( - 384*atan(sqrt(x**2 - x - 1) + x - 1) + 16*sqrt(x**2 - x - 1)*x**2 - 12 
4*sqrt(x**2 - x - 1)*x + 230*sqrt(x**2 - x - 1) + 231*log((2*sqrt(x**2 - x 
 - 1) + 2*x - 1)/sqrt(5)))/48