3.135 \(\int \frac {x^2}{(a+8 x-8 x^2+4 x^3-x^4)^2} \, dx\)

Optimal. Leaf size=225 \[ \frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac {(x-1)^2+1}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\left (a+\sqrt {a+4}+4\right ) \tan ^{-1}\left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{8 (a+3) (a+4) \sqrt {1-\sqrt {a+4}}}-\frac {\left (a-\sqrt {a+4}+4\right ) \tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{8 (a+3) (a+4) \sqrt {\sqrt {a+4}+1}}+\frac {\tanh ^{-1}\left (\frac {(x-1)^2+1}{\sqrt {a+4}}\right )}{2 (a+4)^{3/2}} \]

[Out]

1/2*(1+(-1+x)^2)/(4+a)/(3+a-2*(-1+x)^2-(-1+x)^4)+1/4*(4+a)*(2+(-1+x)^2)*(-1+x)/(a^2+7*a+12)/(3+a-2*(-1+x)^2-(-
1+x)^4)+1/2*arctanh((1+(-1+x)^2)/(4+a)^(1/2))/(4+a)^(3/2)-1/8*arctan((-1+x)/(1-(4+a)^(1/2))^(1/2))*(4+a+(4+a)^
(1/2))/(3+a)/(4+a)/(1-(4+a)^(1/2))^(1/2)-1/8*arctan((-1+x)/(1+(4+a)^(1/2))^(1/2))*(4+a-(4+a)^(1/2))/(3+a)/(4+a
)/(1+(4+a)^(1/2))^(1/2)

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Rubi [A]  time = 0.21, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1680, 1673, 1178, 1166, 204, 12, 1107, 614, 618, 206} \[ \frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac {(x-1)^2+1}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\left (a+\sqrt {a+4}+4\right ) \tan ^{-1}\left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{8 (a+3) (a+4) \sqrt {1-\sqrt {a+4}}}-\frac {\left (a-\sqrt {a+4}+4\right ) \tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{8 (a+3) (a+4) \sqrt {\sqrt {a+4}+1}}+\frac {\tanh ^{-1}\left (\frac {(x-1)^2+1}{\sqrt {a+4}}\right )}{2 (a+4)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2,x]

[Out]

(1 + (-1 + x)^2)/(2*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((4 + a)*(2 + (-1 + x)^2)*(-1 + x))/(4*(12
+ 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((4 + a + Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 - Sqrt[4 + a
]]])/(8*(3 + a)*(4 + a)*Sqrt[1 - Sqrt[4 + a]]) - ((4 + a - Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]]
)/(8*(3 + a)*(4 + a)*Sqrt[1 + Sqrt[4 + a]]) + ArcTanh[(1 + (-1 + x)^2)/Sqrt[4 + a]]/(2*(4 + a)^(3/2))

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 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 614

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

Rule 618

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

Rule 1107

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

Rule 1166

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

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx &=\operatorname {Subst}\left (\int \frac {(1+x)^2}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )\\ &=\operatorname {Subst}\left (\int \frac {2 x}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )+\operatorname {Subst}\left (\int \frac {1+x^2}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )\\ &=\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+2 \operatorname {Subst}\left (\int \frac {x}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )-\frac {\operatorname {Subst}\left (\int \frac {-4 (4+a)-2 (4+a) x^2}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )}{8 \left (12+7 a+a^2\right )}\\ &=\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {\left (4+a-\sqrt {4+a}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {4+a}-x^2} \, dx,x,-1+x\right )}{8 \left (12+7 a+a^2\right )}+\frac {\left (4+a+\sqrt {4+a}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {4+a}-x^2} \, dx,x,-1+x\right )}{8 \left (12+7 a+a^2\right )}+\operatorname {Subst}\left (\int \frac {1}{\left (3+a-2 x-x^2\right )^2} \, dx,x,(-1+x)^2\right )\\ &=\frac {1+(-1+x)^2}{2 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {\left (4+a+\sqrt {4+a}\right ) \tan ^{-1}\left (\frac {1-x}{\sqrt {1-\sqrt {4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt {1-\sqrt {4+a}}}+\frac {\left (4+a-\sqrt {4+a}\right ) \tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt {1+\sqrt {4+a}}}+\frac {\operatorname {Subst}\left (\int \frac {1}{3+a-2 x-x^2} \, dx,x,(-1+x)^2\right )}{2 (4+a)}\\ &=\frac {1+(-1+x)^2}{2 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {\left (4+a+\sqrt {4+a}\right ) \tan ^{-1}\left (\frac {1-x}{\sqrt {1-\sqrt {4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt {1-\sqrt {4+a}}}+\frac {\left (4+a-\sqrt {4+a}\right ) \tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt {1+\sqrt {4+a}}}-\frac {\operatorname {Subst}\left (\int \frac {1}{4 (4+a)-x^2} \, dx,x,-2 \left (1+(-1+x)^2\right )\right )}{4+a}\\ &=\frac {1+(-1+x)^2}{2 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {\left (4+a+\sqrt {4+a}\right ) \tan ^{-1}\left (\frac {1-x}{\sqrt {1-\sqrt {4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt {1-\sqrt {4+a}}}+\frac {\left (4+a-\sqrt {4+a}\right ) \tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt {1+\sqrt {4+a}}}+\frac {\tanh ^{-1}\left (\frac {1+(-1+x)^2}{\sqrt {4+a}}\right )}{2 (4+a)^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 182, normalized size = 0.81 \[ \frac {a \left (x^3-x^2+x+1\right )+2 x \left (2 x^2-3 x+4\right )}{4 (a+3) (a+4) \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )}-\frac {\text {RootSum}\left [-\text {$\#$1}^4+4 \text {$\#$1}^3-8 \text {$\#$1}^2+8 \text {$\#$1}+a\& ,\frac {\text {$\#$1}^2 a \log (x-\text {$\#$1})+4 \text {$\#$1}^2 \log (x-\text {$\#$1})+2 \text {$\#$1} a \log (x-\text {$\#$1})-a \log (x-\text {$\#$1})+4 \text {$\#$1} \log (x-\text {$\#$1})}{\text {$\#$1}^3-3 \text {$\#$1}^2+4 \text {$\#$1}-2}\& \right ]}{16 \left (a^2+7 a+12\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2,x]

[Out]

(2*x*(4 - 3*x + 2*x^2) + a*(1 + x - x^2 + x^3))/(4*(3 + a)*(4 + a)*(a - x*(-8 + 8*x - 4*x^2 + x^3))) - RootSum
[a + 8*#1 - 8*#1^2 + 4*#1^3 - #1^4 & , (-(a*Log[x - #1]) + 4*Log[x - #1]*#1 + 2*a*Log[x - #1]*#1 + 4*Log[x - #
1]*#1^2 + a*Log[x - #1]*#1^2)/(-2 + 4*#1 - 3*#1^2 + #1^3) & ]/(16*(12 + 7*a + a^2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-x^4+4*x^3-8*x^2+a+8*x)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-x^4+4*x^3-8*x^2+a+8*x)^2,x, algorithm="giac")

[Out]

integrate(x^2/(x^4 - 4*x^3 + 8*x^2 - a - 8*x)^2, x)

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maple [C]  time = 0.01, size = 160, normalized size = 0.71 \[ \frac {\left (\left (-a -4\right ) \RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )^{2}+2 \left (-a -2\right ) \RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )+a \right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )+x \right )}{16 \left (a +3\right ) \left (a +4\right ) \left (\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )^{3}-3 \RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )^{2}+4 \RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )-2\right )}+\frac {-\frac {x^{3}}{4 \left (a +3\right )}+\frac {\left (a +6\right ) x^{2}}{4 \left (a +3\right ) \left (a +4\right )}-\frac {a}{4 \left (a +3\right ) \left (a +4\right )}-\frac {\left (a +8\right ) x}{4 \left (a +3\right ) \left (a +4\right )}}{x^{4}-4 x^{3}+8 x^{2}-a -8 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-x^4+4*x^3-8*x^2+a+8*x)^2,x)

[Out]

(-1/4/(a+3)*x^3+1/4*(a+6)/(a+3)/(a+4)*x^2-1/4*(a+8)/(a+3)/(a+4)*x-1/4*a/(a+3)/(a+4))/(x^4-4*x^3+8*x^2-a-8*x)+1
/16/(a+3)/(a+4)*sum(((-a-4)*_R^2+2*(-a-2)*_R+a)/(_R^3-3*_R^2+4*_R-2)*ln(-_R+x),_R=RootOf(_Z^4-4*_Z^3+8*_Z^2-8*
_Z-a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (a + 4\right )} x^{3} - {\left (a + 6\right )} x^{2} + {\left (a + 8\right )} x + a}{4 \, {\left ({\left (a^{2} + 7 \, a + 12\right )} x^{4} - 4 \, {\left (a^{2} + 7 \, a + 12\right )} x^{3} - a^{3} + 8 \, {\left (a^{2} + 7 \, a + 12\right )} x^{2} - 7 \, a^{2} - 8 \, {\left (a^{2} + 7 \, a + 12\right )} x - 12 \, a\right )}} - \frac {\int \frac {{\left (a + 4\right )} x^{2} + 2 \, {\left (a + 2\right )} x - a}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x}}{4 \, {\left (a^{2} + 7 \, a + 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-x^4+4*x^3-8*x^2+a+8*x)^2,x, algorithm="maxima")

[Out]

-1/4*((a + 4)*x^3 - (a + 6)*x^2 + (a + 8)*x + a)/((a^2 + 7*a + 12)*x^4 - 4*(a^2 + 7*a + 12)*x^3 - a^3 + 8*(a^2
+ 7*a + 12)*x^2 - 7*a^2 - 8*(a^2 + 7*a + 12)*x - 12*a) - 1/4*integrate(((a + 4)*x^2 + 2*(a + 2)*x - a)/(x^4 -
4*x^3 + 8*x^2 - a - 8*x), x)/(a^2 + 7*a + 12)

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mupad [B]  time = 2.85, size = 1218, normalized size = 5.41 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2,x)

[Out]

symsum(log((x*(40*a + 7*a^2 + 56))/(8*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576)) - (48*a + 12*a^2 - a^3
)/(64*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576)) - root(12952010752*a^3*z^4 + 31653888*a^7*z^4 + 216268
8*a^8*z^4 + 65536*a^9*z^4 + 18119393280*a*z^4 + 20082327552*a^2*z^4 + 1473773568*a^5*z^4 + 5357174784*a^4*z^4
+ 269680640*a^6*z^4 + 7247757312*z^4 - 24215552*a^2*z^2 - 8986624*a^3*z^2 - 1878016*a^4*z^2 - 209408*a^5*z^2 -
9728*a^6*z^2 - 34865152*a*z^2 - 20971520*z^2 + 237568*a^2*z + 53248*a^3*z + 5888*a^4*z + 256*a^5*z + 524288*a
*z + 458752*z + 1792*a + 1024*a^2 + 144*a^3 - a^4, z, k)*((28160*a + 11328*a^2 + 2064*a^3 + 144*a^4 + 26624)/(
64*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576)) + root(12952010752*a^3*z^4 + 31653888*a^7*z^4 + 2162688*a
^8*z^4 + 65536*a^9*z^4 + 18119393280*a*z^4 + 20082327552*a^2*z^4 + 1473773568*a^5*z^4 + 5357174784*a^4*z^4 + 2
69680640*a^6*z^4 + 7247757312*z^4 - 24215552*a^2*z^2 - 8986624*a^3*z^2 - 1878016*a^4*z^2 - 209408*a^5*z^2 - 97
28*a^6*z^2 - 34865152*a*z^2 - 20971520*z^2 + 237568*a^2*z + 53248*a^3*z + 5888*a^4*z + 256*a^5*z + 524288*a*z
+ 458752*z + 1792*a + 1024*a^2 + 144*a^3 - a^4, z, k)*(root(12952010752*a^3*z^4 + 31653888*a^7*z^4 + 2162688*a
^8*z^4 + 65536*a^9*z^4 + 18119393280*a*z^4 + 20082327552*a^2*z^4 + 1473773568*a^5*z^4 + 5357174784*a^4*z^4 + 2
69680640*a^6*z^4 + 7247757312*z^4 - 24215552*a^2*z^2 - 8986624*a^3*z^2 - 1878016*a^4*z^2 - 209408*a^5*z^2 - 97
28*a^6*z^2 - 34865152*a*z^2 - 20971520*z^2 + 237568*a^2*z + 53248*a^3*z + 5888*a^4*z + 256*a^5*z + 524288*a*z
+ 458752*z + 1792*a + 1024*a^2 + 144*a^3 - a^4, z, k)*((15728640*a + 10878976*a^2 + 3997696*a^3 + 823296*a^4 +
90112*a^5 + 4096*a^6 + 9437184)/(64*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576)) - (x*(1966080*a + 13598
72*a^2 + 499712*a^3 + 102912*a^4 + 11264*a^5 + 512*a^6 + 1179648))/(8*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^
5 + 576))) - (1359872*a + 749568*a^2 + 205824*a^3 + 28160*a^4 + 1536*a^5 + 983040)/(64*(816*a + 460*a^2 + 129*
a^3 + 18*a^4 + a^5 + 576)) + (x*(104448*a + 58880*a^2 + 16512*a^3 + 2304*a^4 + 128*a^5 + 73728))/(8*(816*a + 4
60*a^2 + 129*a^3 + 18*a^4 + a^5 + 576))) + (x*(448*a + 104*a^2 - 2*a^3 - 2*a^4 + 512))/(8*(816*a + 460*a^2 + 1
29*a^3 + 18*a^4 + a^5 + 576))))*root(12952010752*a^3*z^4 + 31653888*a^7*z^4 + 2162688*a^8*z^4 + 65536*a^9*z^4
+ 18119393280*a*z^4 + 20082327552*a^2*z^4 + 1473773568*a^5*z^4 + 5357174784*a^4*z^4 + 269680640*a^6*z^4 + 7247
757312*z^4 - 24215552*a^2*z^2 - 8986624*a^3*z^2 - 1878016*a^4*z^2 - 209408*a^5*z^2 - 9728*a^6*z^2 - 34865152*a
*z^2 - 20971520*z^2 + 237568*a^2*z + 53248*a^3*z + 5888*a^4*z + 256*a^5*z + 524288*a*z + 458752*z + 1792*a + 1
024*a^2 + 144*a^3 - a^4, z, k), k, 1, 4) + (x^3/(4*(a + 3)) + a/(4*(a + 3)*(a + 4)) - (x^2*(a + 6))/(4*(a + 3)
*(a + 4)) + (x*(a + 8))/(4*(a + 3)*(a + 4)))/(a + 8*x - 8*x^2 + 4*x^3 - x^4)

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sympy [B]  time = 43.73, size = 561, normalized size = 2.49 \[ \frac {- a + x^{3} \left (- a - 4\right ) + x^{2} \left (a + 6\right ) + x \left (- a - 8\right )}{- 4 a^{3} - 28 a^{2} - 48 a + x^{4} \left (4 a^{2} + 28 a + 48\right ) + x^{3} \left (- 16 a^{2} - 112 a - 192\right ) + x^{2} \left (32 a^{2} + 224 a + 384\right ) + x \left (- 32 a^{2} - 224 a - 384\right )} + \operatorname {RootSum} {\left (t^{4} \left (65536 a^{9} + 2162688 a^{8} + 31653888 a^{7} + 269680640 a^{6} + 1473773568 a^{5} + 5357174784 a^{4} + 12952010752 a^{3} + 20082327552 a^{2} + 18119393280 a + 7247757312\right ) + t^{2} \left (- 9728 a^{6} - 209408 a^{5} - 1878016 a^{4} - 8986624 a^{3} - 24215552 a^{2} - 34865152 a - 20971520\right ) + t \left (256 a^{5} + 5888 a^{4} + 53248 a^{3} + 237568 a^{2} + 524288 a + 458752\right ) - a^{4} + 144 a^{3} + 1024 a^{2} + 1792 a, \left (t \mapsto t \log {\left (x + \frac {4096 t^{3} a^{12} - 61440 t^{3} a^{11} - 5480448 t^{3} a^{10} - 111403008 t^{3} a^{9} - 1227173888 t^{3} a^{8} - 8682876928 t^{3} a^{7} - 42187440128 t^{3} a^{6} - 144630284288 t^{3} a^{5} - 350972280832 t^{3} a^{4} - 591750234112 t^{3} a^{3} - 660716126208 t^{3} a^{2} - 439848271872 t^{3} a - 132271570944 t^{3} - 28672 t^{2} a^{10} - 993280 t^{2} a^{9} - 15400960 t^{2} a^{8} - 140742656 t^{2} a^{7} - 839462912 t^{2} a^{6} - 3414427648 t^{2} a^{5} - 9590087680 t^{2} a^{4} - 18363547648 t^{2} a^{3} - 22938255360 t^{2} a^{2} - 16873684992 t^{2} a - 5549064192 t^{2} - 848 t a^{9} - 6096 t a^{8} + 174608 t a^{7} + 3323792 t a^{6} + 26276224 t a^{5} + 119009280 t a^{4} + 332017664 t a^{3} + 566497280 t a^{2} + 544112640 t a + 225837056 t + 11 a^{8} + 958 a^{7} + 17419 a^{6} + 142964 a^{5} + 632632 a^{4} + 1567552 a^{3} + 2049792 a^{2} + 1100800 a}{a^{8} + 870 a^{7} + 18289 a^{6} + 165176 a^{5} + 824560 a^{4} + 2452288 a^{3} + 4340224 a^{2} + 4229120 a + 1748992} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(-x**4+4*x**3-8*x**2+a+8*x)**2,x)

[Out]

(-a + x**3*(-a - 4) + x**2*(a + 6) + x*(-a - 8))/(-4*a**3 - 28*a**2 - 48*a + x**4*(4*a**2 + 28*a + 48) + x**3*
(-16*a**2 - 112*a - 192) + x**2*(32*a**2 + 224*a + 384) + x*(-32*a**2 - 224*a - 384)) + RootSum(_t**4*(65536*a
**9 + 2162688*a**8 + 31653888*a**7 + 269680640*a**6 + 1473773568*a**5 + 5357174784*a**4 + 12952010752*a**3 + 2
0082327552*a**2 + 18119393280*a + 7247757312) + _t**2*(-9728*a**6 - 209408*a**5 - 1878016*a**4 - 8986624*a**3
- 24215552*a**2 - 34865152*a - 20971520) + _t*(256*a**5 + 5888*a**4 + 53248*a**3 + 237568*a**2 + 524288*a + 45
8752) - a**4 + 144*a**3 + 1024*a**2 + 1792*a, Lambda(_t, _t*log(x + (4096*_t**3*a**12 - 61440*_t**3*a**11 - 54
80448*_t**3*a**10 - 111403008*_t**3*a**9 - 1227173888*_t**3*a**8 - 8682876928*_t**3*a**7 - 42187440128*_t**3*a
**6 - 144630284288*_t**3*a**5 - 350972280832*_t**3*a**4 - 591750234112*_t**3*a**3 - 660716126208*_t**3*a**2 -
439848271872*_t**3*a - 132271570944*_t**3 - 28672*_t**2*a**10 - 993280*_t**2*a**9 - 15400960*_t**2*a**8 - 1407
42656*_t**2*a**7 - 839462912*_t**2*a**6 - 3414427648*_t**2*a**5 - 9590087680*_t**2*a**4 - 18363547648*_t**2*a*
*3 - 22938255360*_t**2*a**2 - 16873684992*_t**2*a - 5549064192*_t**2 - 848*_t*a**9 - 6096*_t*a**8 + 174608*_t*
a**7 + 3323792*_t*a**6 + 26276224*_t*a**5 + 119009280*_t*a**4 + 332017664*_t*a**3 + 566497280*_t*a**2 + 544112
640*_t*a + 225837056*_t + 11*a**8 + 958*a**7 + 17419*a**6 + 142964*a**5 + 632632*a**4 + 1567552*a**3 + 2049792
*a**2 + 1100800*a)/(a**8 + 870*a**7 + 18289*a**6 + 165176*a**5 + 824560*a**4 + 2452288*a**3 + 4340224*a**2 + 4
229120*a + 1748992))))

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