3.15.6 \(\int \frac {(-3+x)^6 (-1-x+x^2)^{3/2}}{-1+x} \, dx\)

Optimal. Leaf size=101 \[ \frac {19451047 \log \left (2 \sqrt {x^2-x-1}-2 x+1\right )}{65536}+128 \tan ^{-1}\left (\sqrt {x^2-x-1}-x+1\right )+\frac {\sqrt {x^2-x-1} \left (1146880 x^8-23296000 x^7+199009280 x^6-910869760 x^5+2304529024 x^4-2700564848 x^3-508033624 x^2+4423205098 x-1245336401\right )}{10321920} \]

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Rubi [A]  time = 0.37, antiderivative size = 199, normalized size of antiderivative = 1.97, number of steps used = 12, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1653, 814, 843, 621, 206, 724, 204} \begin {gather*} \frac {1}{9} \left (x^2-x-1\right )^{5/2} (1-x)^4+\frac {229}{144} \left (x^2-x-1\right )^{5/2} (1-x)^3+\frac {19927 \left (x^2-x-1\right )^{5/2} (1-x)^2}{2016}+\frac {281233 \left (x^2-x-1\right )^{5/2} (1-x)}{8064}+\frac {6158183 \left (x^2-x-1\right )^{5/2}}{80640}+\frac {(903871-1283454 x) \left (x^2-x-1\right )^{3/2}}{12288}-\frac {(5567931-6941558 x) \sqrt {x^2-x-1}}{32768}-64 \tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )+\frac {19451047 \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )}{65536} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-3 + x)^6*(-1 - x + x^2)^(3/2))/(-1 + x),x]

[Out]

-1/32768*((5567931 - 6941558*x)*Sqrt[-1 - x + x^2]) + ((903871 - 1283454*x)*(-1 - x + x^2)^(3/2))/12288 + (615
8183*(-1 - x + x^2)^(5/2))/80640 + (281233*(1 - x)*(-1 - x + x^2)^(5/2))/8064 + (19927*(1 - x)^2*(-1 - x + x^2
)^(5/2))/2016 + (229*(1 - x)^3*(-1 - x + x^2)^(5/2))/144 + ((1 - x)^4*(-1 - x + x^2)^(5/2))/9 - 64*ArcTan[(3 -
x)/(2*Sqrt[-1 - x + x^2])] + (19451047*ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])])/65536

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]))

Rubi steps

\begin {align*} \int \frac {(-3+x)^6 \left (-1-x+x^2\right )^{3/2}}{-1+x} \, dx &=\frac {1}{9} (1-x)^4 \left (-1-x+x^2\right )^{5/2}+\frac {1}{9} \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\\ &=\frac {229}{144} (1-x)^3 \left (-1-x+x^2\right )^{5/2}+\frac {1}{9} (1-x)^4 \left (-1-x+x^2\right )^{5/2}+\frac {1}{72} \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\\ &=\frac {19927 (1-x)^2 \left (-1-x+x^2\right )^{5/2}}{2016}+\frac {229}{144} (1-x)^3 \left (-1-x+x^2\right )^{5/2}+\frac {1}{9} (1-x)^4 \left (-1-x+x^2\right )^{5/2}+\frac {1}{504} \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\\ &=\frac {281233 (1-x) \left (-1-x+x^2\right )^{5/2}}{8064}+\frac {19927 (1-x)^2 \left (-1-x+x^2\right )^{5/2}}{2016}+\frac {229}{144} (1-x)^3 \left (-1-x+x^2\right )^{5/2}+\frac {1}{9} (1-x)^4 \left (-1-x+x^2\right )^{5/2}+\frac {\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}\\ &=\frac {6158183 \left (-1-x+x^2\right )^{5/2}}{80640}+\frac {281233 (1-x) \left (-1-x+x^2\right )^{5/2}}{8064}+\frac {19927 (1-x)^2 \left (-1-x+x^2\right )^{5/2}}{2016}+\frac {229}{144} (1-x)^3 \left (-1-x+x^2\right )^{5/2}+\frac {1}{9} (1-x)^4 \left (-1-x+x^2\right )^{5/2}+\frac {\int \frac {\left (\frac {233109765}{32}-\frac {202144005 x}{32}\right ) \left (-1-x+x^2\right )^{3/2}}{-1+x} \, dx}{15120}\\ &=\frac {(903871-1283454 x) \left (-1-x+x^2\right )^{3/2}}{12288}+\frac {6158183 \left (-1-x+x^2\right )^{5/2}}{80640}+\frac {281233 (1-x) \left (-1-x+x^2\right )^{5/2}}{8064}+\frac {19927 (1-x)^2 \left (-1-x+x^2\right )^{5/2}}{2016}+\frac {229}{144} (1-x)^3 \left (-1-x+x^2\right )^{5/2}+\frac {1}{9} (1-x)^4 \left (-1-x+x^2\right )^{5/2}-\frac {\int \frac {\left (\frac {3775338315}{64}-\frac {3279886155 x}{64}\right ) \sqrt {-1-x+x^2}}{-1+x} \, dx}{120960}\\ &=-\frac {(5567931-6941558 x) \sqrt {-1-x+x^2}}{32768}+\frac {(903871-1283454 x) \left (-1-x+x^2\right )^{3/2}}{12288}+\frac {6158183 \left (-1-x+x^2\right )^{5/2}}{80640}+\frac {281233 (1-x) \left (-1-x+x^2\right )^{5/2}}{8064}+\frac {19927 (1-x)^2 \left (-1-x+x^2\right )^{5/2}}{2016}+\frac {229}{144} (1-x)^3 \left (-1-x+x^2\right )^{5/2}+\frac {1}{9} (1-x)^4 \left (-1-x+x^2\right )^{5/2}+\frac {\int \frac {\frac {22344856695}{128}-\frac {18381239415 x}{128}}{(-1+x) \sqrt {-1-x+x^2}} \, dx}{483840}\\ &=-\frac {(5567931-6941558 x) \sqrt {-1-x+x^2}}{32768}+\frac {(903871-1283454 x) \left (-1-x+x^2\right )^{3/2}}{12288}+\frac {6158183 \left (-1-x+x^2\right )^{5/2}}{80640}+\frac {281233 (1-x) \left (-1-x+x^2\right )^{5/2}}{8064}+\frac {19927 (1-x)^2 \left (-1-x+x^2\right )^{5/2}}{2016}+\frac {229}{144} (1-x)^3 \left (-1-x+x^2\right )^{5/2}+\frac {1}{9} (1-x)^4 \left (-1-x+x^2\right )^{5/2}+64 \int \frac {1}{(-1+x) \sqrt {-1-x+x^2}} \, dx-\frac {19451047 \int \frac {1}{\sqrt {-1-x+x^2}} \, dx}{65536}\\ &=-\frac {(5567931-6941558 x) \sqrt {-1-x+x^2}}{32768}+\frac {(903871-1283454 x) \left (-1-x+x^2\right )^{3/2}}{12288}+\frac {6158183 \left (-1-x+x^2\right )^{5/2}}{80640}+\frac {281233 (1-x) \left (-1-x+x^2\right )^{5/2}}{8064}+\frac {19927 (1-x)^2 \left (-1-x+x^2\right )^{5/2}}{2016}+\frac {229}{144} (1-x)^3 \left (-1-x+x^2\right )^{5/2}+\frac {1}{9} (1-x)^4 \left (-1-x+x^2\right )^{5/2}-128 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3+x}{\sqrt {-1-x+x^2}}\right )-\frac {19451047 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x}{\sqrt {-1-x+x^2}}\right )}{32768}\\ &=-\frac {(5567931-6941558 x) \sqrt {-1-x+x^2}}{32768}+\frac {(903871-1283454 x) \left (-1-x+x^2\right )^{3/2}}{12288}+\frac {6158183 \left (-1-x+x^2\right )^{5/2}}{80640}+\frac {281233 (1-x) \left (-1-x+x^2\right )^{5/2}}{8064}+\frac {19927 (1-x)^2 \left (-1-x+x^2\right )^{5/2}}{2016}+\frac {229}{144} (1-x)^3 \left (-1-x+x^2\right )^{5/2}+\frac {1}{9} (1-x)^4 \left (-1-x+x^2\right )^{5/2}-64 \tan ^{-1}\left (\frac {3-x}{2 \sqrt {-1-x+x^2}}\right )+\frac {19451047 \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )}{65536}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 107, normalized size = 1.06 \begin {gather*} -64 \tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )+\frac {19451047 \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )}{65536}+\frac {\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 )}{10321920} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-3 + x)^6*(-1 - x + x^2)^(3/2))/(-1 + x),x]

[Out]

(Sqrt[-1 - x + x^2]*(-1245336401 + 4423205098*x - 508033624*x^2 - 2700564848*x^3 + 2304529024*x^4 - 910869760*
x^5 + 199009280*x^6 - 23296000*x^7 + 1146880*x^8))/10321920 - 64*ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])] + (194
51047*ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])])/65536

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IntegrateAlgebraic [A]  time = 0.52, size = 101, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1-x+x^2} \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 )}{10321920}+128 \tan ^{-1}\left (1-x+\sqrt {-1-x+x^2}\right )+\frac {19451047 \log \left (1-2 x+2 \sqrt {-1-x+x^2}\right )}{65536} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-3 + x)^6*(-1 - x + x^2)^(3/2))/(-1 + x),x]

[Out]

(Sqrt[-1 - x + x^2]*(-1245336401 + 4423205098*x - 508033624*x^2 - 2700564848*x^3 + 2304529024*x^4 - 910869760*
x^5 + 199009280*x^6 - 23296000*x^7 + 1146880*x^8))/10321920 + 128*ArcTan[1 - x + Sqrt[-1 - x + x^2]] + (194510
47*Log[1 - 2*x + 2*Sqrt[-1 - x + x^2]])/65536

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fricas [A]  time = 0.57, size = 91, normalized size = 0.90 \begin {gather*} \frac {1}{10321920} \, {\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 )} \sqrt {x^{2} - x - 1} + 128 \, \arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) + \frac {19451047}{65536} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+x)^6*(x^2-x-1)^(3/2)/(-1+x),x, algorithm="fricas")

[Out]

1/10321920*(1146880*x^8 - 23296000*x^7 + 199009280*x^6 - 910869760*x^5 + 2304529024*x^4 - 2700564848*x^3 - 508
033624*x^2 + 4423205098*x - 1245336401)*sqrt(x^2 - x - 1) + 128*arctan(-x + sqrt(x^2 - x - 1) + 1) + 19451047/
65536*log(-2*x + 2*sqrt(x^2 - x - 1) + 1)

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giac [A]  time = 0.18, size = 92, normalized size = 0.91 \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((-3+x)^6*(x^2-x-1)^(3/2)/(-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 = 0.36, size = 88, normalized size = 0.87

method	result	size
risch	\(\frac {\sqrt {x^{2}-x -1}\, \left (1146880 x^{8}-23296000 x^{7}[phone] x^{6}-910869760 x^{5}[phone] x^{4}-2700564848 x^{3}-508033624 x^{2}[phone] x -1245336401\right )}{10321920}-\frac {19451047 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x -1}\right )}{65536}+64 \arctan \left (\frac {-3+x}{2 \sqrt {\left (-1+x \right )^{2}-2+x}}\right )\)	\(88\)
trager	\(\left (\frac {1}{9} x^{8}-\frac {325}{144} x^{7}+\frac {38869}{2016} x^{6}-\frac {711617}{8064} x^{5}+\frac {2572019}{11520} x^{4}-\frac {168785303}{645120} x^{3}-\frac {9072029}{184320} x^{2}+\frac {2211602549}{5160960} x -\frac {1245336401}{10321920}\right ) \sqrt {x^{2}-x -1}+\frac {19451047 \ln \left (1-2 x +2 \sqrt {x^{2}-x -1}\right )}{65536}+64 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x -3 \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \sqrt {x^{2}-x -1}}{-1+x}\right )\)	\(117\)
default	\(8 \left (-1+2 x \right ) \sqrt {\left (-1+x \right )^{2}-2+x}+\frac {x^{4} \left (x^{2}-x -1\right )^{\frac {5}{2}}}{9}-\frac {293 x^{3} \left (x^{2}-x -1\right )^{\frac {5}{2}}}{144}+\frac {30889 x^{2} \left (x^{2}-x -1\right )^{\frac {5}{2}}}{2016}-\frac {482705 x \left (x^{2}-x -1\right )^{\frac {5}{2}}}{8064}-\frac {213909 \left (-1+2 x \right ) \left (x^{2}-x -1\right )^{\frac {3}{2}}}{4096}+\frac {3208635 \sqrt {x^{2}-x -1}\, \left (-1+2 x \right )}{32768}+64 \arctan \left (\frac {-3+x}{2 \sqrt {\left (-1+x \right )^{2}-2+x}}\right )+\frac {64 \left (\left (-1+x \right )^{2}-2+x \right )^{\frac {3}{2}}}{3}-64 \sqrt {\left (-1+x \right )^{2}-2+x}-52 \ln \left (-\frac {1}{2}+x +\sqrt {\left (-1+x \right )^{2}-2+x}\right )-\frac {16043175 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x -1}\right )}{65536}+\frac {9904793 \left (x^{2}-x -1\right )^{\frac {5}{2}}}{80640}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3+x)^6*(x^2-x-1)^(3/2)/(-1+x),x,method=_RETURNVERBOSE)

[Out]

1/10321920*(x^2-x-1)^(1/2)*(1146880*x^8-23296000*x^7+199009280*x^6-910869760*x^5+2304529024*x^4-2700564848*x^3
-508033624*x^2+4423205098*x-1245336401)-19451047/65536*ln(x-1/2+(x^2-x-1)^(1/2))+64*arctan(1/2*(-3+x)/((-1+x)^
2-2+x)^(1/2))

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maxima [A]  time = 0.42, size = 168, normalized size = 1.66 \begin {gather*} \frac {1}{9} \, {\left (x^{2} - x - 1\right )}^{\frac {5}{2}} x^{4} - \frac {293}{144} \, {\left (x^{2} - x - 1\right )}^{\frac {5}{2}} x^{3} + \frac {30889}{2016} \, {\left (x^{2} - x - 1\right )}^{\frac {5}{2}} x^{2} - \frac {482705}{8064} \, {\left (x^{2} - x - 1\right )}^{\frac {5}{2}} x + \frac {9904793}{80640} \, {\left (x^{2} - x - 1\right )}^{\frac {5}{2}} - \frac {213909}{2048} \, {\left (x^{2} - x - 1\right )}^{\frac {3}{2}} x + \frac {903871}{12288} \, {\left (x^{2} - x - 1\right )}^{\frac {3}{2}} + \frac {3470779}{16384} \, \sqrt {x^{2} - x - 1} x - \frac {5567931}{32768} \, \sqrt {x^{2} - x - 1} + 64 \, \arcsin \left (\frac {\sqrt {5} x}{5 \, {\left | x - 1 \right |}} - \frac {3 \, \sqrt {5}}{5 \, {\left | x - 1 \right |}}\right ) - \frac {19451047}{65536} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - x - 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+x)^6*(x^2-x-1)^(3/2)/(-1+x),x, algorithm="maxima")

[Out]

1/9*(x^2 - x - 1)^(5/2)*x^4 - 293/144*(x^2 - x - 1)^(5/2)*x^3 + 30889/2016*(x^2 - x - 1)^(5/2)*x^2 - 482705/80
64*(x^2 - x - 1)^(5/2)*x + 9904793/80640*(x^2 - x - 1)^(5/2) - 213909/2048*(x^2 - x - 1)^(3/2)*x + 903871/1228
8*(x^2 - x - 1)^(3/2) + 3470779/16384*sqrt(x^2 - x - 1)*x - 5567931/32768*sqrt(x^2 - x - 1) + 64*arcsin(1/5*sq
rt(5)*x/abs(x - 1) - 3/5*sqrt(5)/abs(x - 1)) - 19451047/65536*log(2*x + 2*sqrt(x^2 - x - 1) - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x-3\right )}^6\,{\left (x^2-x-1\right )}^{3/2}}{x-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x - 3)^6*(x^2 - x - 1)^(3/2))/(x - 1),x)

[Out]

int(((x - 3)^6*(x^2 - x - 1)^(3/2))/(x - 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 3\right )^{6} \left (x^{2} - x - 1\right )^{\frac {3}{2}}}{x - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x - 3)**6*(x**2 - x - 1)**(3/2)/(x - 1), x)

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