3.21.24 \(\int \frac {1}{(-1+x) (-2 x^2-3 x^3+x^4)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ \frac {\sqrt {x^4-3 x^3-2 x^2} \left (-453 x^3+1555 x^2+238 x-136\right )}{544 x^3 \left (x^2-3 x-2\right )}+\frac {1}{4} \tan ^{-1}\left (\frac {\frac {x^2}{2}-\frac {1}{2} \sqrt {x^4-3 x^3-2 x^2}-\frac {x}{2}}{x}\right )-\frac {119 \tan ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}-\frac {\sqrt {x^4-3 x^3-2 x^2}}{\sqrt {2}}}{x}\right )}{32 \sqrt {2}} \]

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Rubi [B]  time = 0.23, antiderivative size = 303, normalized size of antiderivative = 2.10, number of steps used = 20, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2056, 960, 740, 12, 724, 204, 834, 806} \begin {gather*} \frac {x (13-3 x)}{17 \sqrt {x^4-3 x^3-2 x^2}}+\frac {13-3 x}{17 x \sqrt {x^4-3 x^3-2 x^2}}+\frac {13-3 x}{17 \sqrt {x^4-3 x^3-2 x^2}}-\frac {(10-x) x}{34 \sqrt {x^4-3 x^3-2 x^2}}-\frac {69 \left (-x^2+3 x+2\right )}{136 x \sqrt {x^4-3 x^3-2 x^2}}+\frac {373 \left (-x^2+3 x+2\right )}{544 \sqrt {x^4-3 x^3-2 x^2}}+\frac {x \sqrt {x^2-3 x-2} \tan ^{-1}\left (\frac {x+7}{4 \sqrt {x^2-3 x-2}}\right )}{8 \sqrt {x^4-3 x^3-2 x^2}}-\frac {119 x \sqrt {x^2-3 x-2} \tan ^{-1}\left (\frac {3 x+4}{2 \sqrt {2} \sqrt {x^2-3 x-2}}\right )}{64 \sqrt {2} \sqrt {x^4-3 x^3-2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((-1 + x)*(-2*x^2 - 3*x^3 + x^4)^(3/2)),x]

[Out]

(13 - 3*x)/(17*Sqrt[-2*x^2 - 3*x^3 + x^4]) + (13 - 3*x)/(17*x*Sqrt[-2*x^2 - 3*x^3 + x^4]) + ((13 - 3*x)*x)/(17
*Sqrt[-2*x^2 - 3*x^3 + x^4]) - ((10 - x)*x)/(34*Sqrt[-2*x^2 - 3*x^3 + x^4]) + (373*(2 + 3*x - x^2))/(544*Sqrt[
-2*x^2 - 3*x^3 + x^4]) - (69*(2 + 3*x - x^2))/(136*x*Sqrt[-2*x^2 - 3*x^3 + x^4]) + (x*Sqrt[-2 - 3*x + x^2]*Arc
Tan[(7 + x)/(4*Sqrt[-2 - 3*x + x^2])])/(8*Sqrt[-2*x^2 - 3*x^3 + x^4]) - (119*x*Sqrt[-2 - 3*x + x^2]*ArcTan[(4
+ 3*x)/(2*Sqrt[2]*Sqrt[-2 - 3*x + x^2])])/(64*Sqrt[2]*Sqrt[-2*x^2 - 3*x^3 + x^4])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 960

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

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps

\begin {align*} \int \frac {1}{(-1+x) \left (-2 x^2-3 x^3+x^4\right )^{3/2}} \, dx &=\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \frac {1}{(-1+x) x^3 \left (-2-3 x+x^2\right )^{3/2}} \, dx}{\sqrt {-2 x^2-3 x^3+x^4}}\\ &=\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \left (\frac {1}{(-1+x) \left (-2-3 x+x^2\right )^{3/2}}-\frac {1}{x^3 \left (-2-3 x+x^2\right )^{3/2}}-\frac {1}{x^2 \left (-2-3 x+x^2\right )^{3/2}}-\frac {1}{x \left (-2-3 x+x^2\right )^{3/2}}\right ) \, dx}{\sqrt {-2 x^2-3 x^3+x^4}}\\ &=\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \frac {1}{(-1+x) \left (-2-3 x+x^2\right )^{3/2}} \, dx}{\sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \frac {1}{x^3 \left (-2-3 x+x^2\right )^{3/2}} \, dx}{\sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \frac {1}{x^2 \left (-2-3 x+x^2\right )^{3/2}} \, dx}{\sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \frac {1}{x \left (-2-3 x+x^2\right )^{3/2}} \, dx}{\sqrt {-2 x^2-3 x^3+x^4}}\\ &=\frac {13-3 x}{17 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {13-3 x}{17 x \sqrt {-2 x^2-3 x^3+x^4}}+\frac {(13-3 x) x}{17 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {(10-x) x}{34 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int -\frac {17}{2 (-1+x) \sqrt {-2-3 x+x^2}} \, dx}{34 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int -\frac {17}{2 x \sqrt {-2-3 x+x^2}} \, dx}{17 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \frac {-\frac {43}{2}+3 x}{x^2 \sqrt {-2-3 x+x^2}} \, dx}{17 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \frac {-\frac {69}{2}+6 x}{x^3 \sqrt {-2-3 x+x^2}} \, dx}{17 \sqrt {-2 x^2-3 x^3+x^4}}\\ &=\frac {13-3 x}{17 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {13-3 x}{17 x \sqrt {-2 x^2-3 x^3+x^4}}+\frac {(13-3 x) x}{17 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {(10-x) x}{34 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {43 \left (2+3 x-x^2\right )}{68 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {69 \left (2+3 x-x^2\right )}{136 x \sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \frac {\frac {717}{4}-\frac {69 x}{2}}{x^2 \sqrt {-2-3 x+x^2}} \, dx}{68 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \frac {1}{(-1+x) \sqrt {-2-3 x+x^2}} \, dx}{4 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \int \frac {1}{x \sqrt {-2-3 x+x^2}} \, dx}{2 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (9 x \sqrt {-2-3 x+x^2}\right ) \int \frac {1}{x \sqrt {-2-3 x+x^2}} \, dx}{8 \sqrt {-2 x^2-3 x^3+x^4}}\\ &=\frac {13-3 x}{17 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {13-3 x}{17 x \sqrt {-2 x^2-3 x^3+x^4}}+\frac {(13-3 x) x}{17 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {(10-x) x}{34 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {373 \left (2+3 x-x^2\right )}{544 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {69 \left (2+3 x-x^2\right )}{136 x \sqrt {-2 x^2-3 x^3+x^4}}+\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-16-x^2} \, dx,x,\frac {-7-x}{\sqrt {-2-3 x+x^2}}\right )}{2 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (x \sqrt {-2-3 x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4-3 x}{\sqrt {-2-3 x+x^2}}\right )}{\sqrt {-2 x^2-3 x^3+x^4}}+\frac {\left (9 x \sqrt {-2-3 x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4-3 x}{\sqrt {-2-3 x+x^2}}\right )}{4 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {\left (159 x \sqrt {-2-3 x+x^2}\right ) \int \frac {1}{x \sqrt {-2-3 x+x^2}} \, dx}{64 \sqrt {-2 x^2-3 x^3+x^4}}\\ &=\frac {13-3 x}{17 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {13-3 x}{17 x \sqrt {-2 x^2-3 x^3+x^4}}+\frac {(13-3 x) x}{17 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {(10-x) x}{34 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {373 \left (2+3 x-x^2\right )}{544 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {69 \left (2+3 x-x^2\right )}{136 x \sqrt {-2 x^2-3 x^3+x^4}}+\frac {x \sqrt {-2-3 x+x^2} \tan ^{-1}\left (\frac {7+x}{4 \sqrt {-2-3 x+x^2}}\right )}{8 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {5 x \sqrt {-2-3 x+x^2} \tan ^{-1}\left (\frac {4+3 x}{2 \sqrt {2} \sqrt {-2-3 x+x^2}}\right )}{8 \sqrt {2} \sqrt {-2 x^2-3 x^3+x^4}}-\frac {\left (159 x \sqrt {-2-3 x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4-3 x}{\sqrt {-2-3 x+x^2}}\right )}{32 \sqrt {-2 x^2-3 x^3+x^4}}\\ &=\frac {13-3 x}{17 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {13-3 x}{17 x \sqrt {-2 x^2-3 x^3+x^4}}+\frac {(13-3 x) x}{17 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {(10-x) x}{34 \sqrt {-2 x^2-3 x^3+x^4}}+\frac {373 \left (2+3 x-x^2\right )}{544 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {69 \left (2+3 x-x^2\right )}{136 x \sqrt {-2 x^2-3 x^3+x^4}}+\frac {x \sqrt {-2-3 x+x^2} \tan ^{-1}\left (\frac {7+x}{4 \sqrt {-2-3 x+x^2}}\right )}{8 \sqrt {-2 x^2-3 x^3+x^4}}-\frac {119 x \sqrt {-2-3 x+x^2} \tan ^{-1}\left (\frac {4+3 x}{2 \sqrt {2} \sqrt {-2-3 x+x^2}}\right )}{64 \sqrt {2} \sqrt {-2 x^2-3 x^3+x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 126, normalized size = 0.88 \begin {gather*} \frac {-1812 x^3+6220 x^2+2023 \sqrt {2} \sqrt {x^2-3 x-2} x^2 \tan ^{-1}\left (\frac {-3 x-4}{2 \sqrt {2} \sqrt {x^2-3 x-2}}\right )-272 \sqrt {x^2-3 x-2} x^2 \tan ^{-1}\left (\frac {-x-7}{4 \sqrt {x^2-3 x-2}}\right )+952 x-544}{2176 x \sqrt {x^2 \left (x^2-3 x-2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((-1 + x)*(-2*x^2 - 3*x^3 + x^4)^(3/2)),x]

[Out]

(-544 + 952*x + 6220*x^2 - 1812*x^3 + 2023*Sqrt[2]*x^2*Sqrt[-2 - 3*x + x^2]*ArcTan[(-4 - 3*x)/(2*Sqrt[2]*Sqrt[
-2 - 3*x + x^2])] - 272*x^2*Sqrt[-2 - 3*x + x^2]*ArcTan[(-7 - x)/(4*Sqrt[-2 - 3*x + x^2])])/(2176*x*Sqrt[x^2*(
-2 - 3*x + x^2)])

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IntegrateAlgebraic [A]  time = 1.30, size = 144, normalized size = 1.00 \begin {gather*} \frac {\left (-136+238 x+1555 x^2-453 x^3\right ) \sqrt {-2 x^2-3 x^3+x^4}}{544 x^3 \left (-2-3 x+x^2\right )}+\frac {1}{4} \tan ^{-1}\left (\frac {-\frac {x}{2}+\frac {x^2}{2}-\frac {1}{2} \sqrt {-2 x^2-3 x^3+x^4}}{x}\right )-\frac {119 \tan ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}-\frac {\sqrt {-2 x^2-3 x^3+x^4}}{\sqrt {2}}}{x}\right )}{32 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((-1 + x)*(-2*x^2 - 3*x^3 + x^4)^(3/2)),x]

[Out]

((-136 + 238*x + 1555*x^2 - 453*x^3)*Sqrt[-2*x^2 - 3*x^3 + x^4])/(544*x^3*(-2 - 3*x + x^2)) + ArcTan[(-1/2*x +
x^2/2 - Sqrt[-2*x^2 - 3*x^3 + x^4]/2)/x]/4 - (119*ArcTan[(x^2/Sqrt[2] - Sqrt[-2*x^2 - 3*x^3 + x^4]/Sqrt[2])/x
])/(32*Sqrt[2])

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fricas [A]  time = 0.47, size = 168, normalized size = 1.17 \begin {gather*} -\frac {906 \, x^{5} - 2718 \, x^{4} - 1812 \, x^{3} - 2023 \, \sqrt {2} {\left (x^{5} - 3 \, x^{4} - 2 \, x^{3}\right )} \arctan \left (-\frac {\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} - 3 \, x^{3} - 2 \, x^{2}}}{2 \, x}\right ) + 272 \, {\left (x^{5} - 3 \, x^{4} - 2 \, x^{3}\right )} \arctan \left (-\frac {x^{2} - x - \sqrt {x^{4} - 3 \, x^{3} - 2 \, x^{2}}}{2 \, x}\right ) + 2 \, \sqrt {x^{4} - 3 \, x^{3} - 2 \, x^{2}} {\left (453 \, x^{3} - 1555 \, x^{2} - 238 \, x + 136\right )}}{1088 \, {\left (x^{5} - 3 \, x^{4} - 2 \, x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)/(x^4-3*x^3-2*x^2)^(3/2),x, algorithm="fricas")

[Out]

-1/1088*(906*x^5 - 2718*x^4 - 1812*x^3 - 2023*sqrt(2)*(x^5 - 3*x^4 - 2*x^3)*arctan(-1/2*(sqrt(2)*x^2 - sqrt(2)
*sqrt(x^4 - 3*x^3 - 2*x^2))/x) + 272*(x^5 - 3*x^4 - 2*x^3)*arctan(-1/2*(x^2 - x - sqrt(x^4 - 3*x^3 - 2*x^2))/x
) + 2*sqrt(x^4 - 3*x^3 - 2*x^2)*(453*x^3 - 1555*x^2 - 238*x + 136))/(x^5 - 3*x^4 - 2*x^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)/(x^4-3*x^3-2*x^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.35, size = 111, normalized size = 0.77

method	result	size
risch	\(-\frac {453 x^{3}-1555 x^{2}-238 x +136}{544 x \sqrt {x^{2} \left (x^{2}-3 x -2\right )}}+\frac {\left (-\frac {\arctan \left (\frac {-7-x}{4 \sqrt {\left (-1+x \right )^{2}-3-x}}\right )}{8}+\frac {119 \sqrt {2}\, \arctan \left (\frac {\left (-4-3 x \right ) \sqrt {2}}{4 \sqrt {x^{2}-3 x -2}}\right )}{128}\right ) x \sqrt {x^{2}-3 x -2}}{\sqrt {x^{2} \left (x^{2}-3 x -2\right )}}\)	\(111\)
default	\(-\frac {x \left (x^{2}-3 x -2\right ) \left (2023 \sqrt {2}\, \arctan \left (\frac {\left (4+3 x \right ) \sqrt {2}}{4 \sqrt {x^{2}-3 x -2}}\right ) x^{2} \sqrt {x^{2}-3 x -2}-272 \arctan \left (\frac {7+x}{4 \sqrt {x^{2}-3 x -2}}\right ) x^{2} \sqrt {x^{2}-3 x -2}+1812 x^{3}-6220 x^{2}-952 x +544\right )}{2176 \left (x^{4}-3 x^{3}-2 x^{2}\right )^{\frac {3}{2}}}\)	\(113\)
trager	\(-\frac {\left (453 x^{3}-1555 x^{2}-238 x +136\right ) \sqrt {x^{4}-3 x^{3}-2 x^{2}}}{544 \left (x^{2}-3 x -2\right ) x^{3}}+\frac {119 \RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {x^{4}-3 x^{3}-2 x^{2}}}{x^{2}}\right )}{128}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+7 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +4 \sqrt {x^{4}-3 x^{3}-2 x^{2}}}{x \left (-1+x \right )}\right )}{8}\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-1+x)/(x^4-3*x^3-2*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/544*(453*x^3-1555*x^2-238*x+136)/x/(x^2*(x^2-3*x-2))^(1/2)+(-1/8*arctan(1/4*(-7-x)/((-1+x)^2-3-x)^(1/2))+11
9/128*2^(1/2)*arctan(1/4*(-4-3*x)*2^(1/2)/(x^2-3*x-2)^(1/2)))*x*(x^2-3*x-2)^(1/2)/(x^2*(x^2-3*x-2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)/(x^4-3*x^3-2*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((x^4 - 3*x^3 - 2*x^2)^(3/2)*(x - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x-1\right )\,{\left (x^4-3\,x^3-2\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x - 1)*(x^4 - 3*x^3 - 2*x^2)^(3/2)),x)

[Out]

int(1/((x - 1)*(x^4 - 3*x^3 - 2*x^2)^(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)/(x**4-3*x**3-2*x**2)**(3/2),x)

[Out]

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

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