Integrand size = 26, antiderivative size = 225 \[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\frac {1+(-1+x)^2}{2 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}-\frac {\left (4+a+\sqrt {4+a}\right ) \arctan \left (\frac {-1+x}{\sqrt {1-\sqrt {4+a}}}\right )}{8 (3+a) (4+a) \sqrt {1-\sqrt {4+a}}}-\frac {\left (4+a-\sqrt {4+a}\right ) \arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )}{8 (3+a) (4+a) \sqrt {1+\sqrt {4+a}}}+\frac {\text {arctanh}\left (\frac {1+(-1+x)^2}{\sqrt {4+a}}\right )}{2 (4+a)^{3/2}} \]
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)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\frac {2 x \left (4-3 x+2 x^2\right )+a \left (1+x-x^2+x^3\right )}{4 (3+a) (4+a) \left (a-x \left (-8+8 x-4 x^2+x^3\right )\right )}-\frac {\text {RootSum}\left [a+8 \text {$\#$1}-8 \text {$\#$1}^2+4 \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-a \log (x-\text {$\#$1})+4 \log (x-\text {$\#$1}) \text {$\#$1}+2 a \log (x-\text {$\#$1}) \text {$\#$1}+4 \log (x-\text {$\#$1}) \text {$\#$1}^2+a \log (x-\text {$\#$1}) \text {$\#$1}^2}{-2+4 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]}{16 \left (12+7 a+a^2\right )} \]
(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))
Time = 0.46 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2459, 2006, 2202, 27, 1432, 1086, 1083, 219, 1492, 27, 1480, 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.
| \(\displaystyle \int \frac {x^2}{\left (a-x^4+4 x^3-8 x^2+8 x\right )^2} \, dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \frac {(x-1)^2+2 (x-1)+1}{\left (a-(x-1)^4-2 (x-1)^2+3\right )^2}d(x-1)\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {x^2}{\left (a-(x-1)^4-2 (x-1)^2+3\right )^2}d(x-1)\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)+\int \frac {2 (x-1)}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)+2 \int \frac {x-1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle \int \frac {1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)^2+\int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle \int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)+\frac {\int \frac {1}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)^2}{2 (a+4)}+\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\int \frac {1}{4 (a+4)-(x-1)^4}d\left (-2 (x-1)^2-2\right )}{a+4}+\int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)+\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}+\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\int -\frac {2 (a+4) \left ((x-1)^2+2\right )}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{8 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}+\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+4) \int \frac {(x-1)^2+2}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{4 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}+\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {(a+4) \left (\frac {1}{2} \left (1-\frac {1}{\sqrt {a+4}}\right ) \int \frac {1}{-(x-1)^2-\sqrt {a+4}-1}d(x-1)+\frac {1}{2} \left (\frac {1}{\sqrt {a+4}}+1\right ) \int \frac {1}{-(x-1)^2+\sqrt {a+4}-1}d(x-1)\right )}{4 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}+\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(a+4) \left (-\frac {\left (\frac {1}{\sqrt {a+4}}+1\right ) \arctan \left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{2 \sqrt {1-\sqrt {a+4}}}-\frac {\left (1-\frac {1}{\sqrt {a+4}}\right ) \arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{2 \sqrt {\sqrt {a+4}+1}}\right )}{4 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}+\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
((4 + a)*(2 + (-1 + x)^2)*(-1 + x))/(4*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x )^2 - (-1 + x)^4)) + x/(2*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ( (4 + a)*(-1/2*((1 + 1/Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 - Sqrt[4 + a]]]) /Sqrt[1 - Sqrt[4 + a]] - ((1 - 1/Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 + Sqr t[4 + a]]])/(2*Sqrt[1 + Sqrt[4 + a]])))/(4*(12 + 7*a + a^2)) - ArcTanh[(-2 - 2*(-1 + x)^2)/(2*Sqrt[4 + a])]/(2*(4 + a)^(3/2))
3.2.35.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
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] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(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]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> 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] + Simp[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]
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(\frac {\frac {x^{3}}{4 a +12}-\frac {\left (6+a \right ) x^{2}}{4 \left (3+a \right ) \left (4+a \right )}+\frac {\left (a +8\right ) x}{4 \left (3+a \right ) \left (4+a \right )}+\frac {a}{4 \left (4+a \right ) \left (3+a \right )}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (\textit {\_R}^{2} \left (4+a \right )+2 \left (a +2\right ) \textit {\_R} -a \right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}}{16 \left (4+a \right ) \left (3+a \right )}\) | \(160\) |
| risch | \(\frac {\frac {x^{3}}{4 a +12}-\frac {\left (6+a \right ) x^{2}}{4 \left (3+a \right ) \left (4+a \right )}+\frac {\left (a +8\right ) x}{4 \left (3+a \right ) \left (4+a \right )}+\frac {a}{4 \left (4+a \right ) \left (3+a \right )}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (\frac {\textit {\_R}^{2}}{3+a}+\frac {2 \left (a +2\right ) \textit {\_R}}{\left (3+a \right ) \left (4+a \right )}-\frac {a}{\left (4+a \right ) \left (3+a \right )}\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{16}\) | \(172\) |
(1/4/(3+a)*x^3-1/4*(6+a)/(3+a)/(4+a)*x^2+1/4*(a+8)/(3+a)/(4+a)*x+1/4*a/(4+ a)/(3+a))/(-x^4+4*x^3-8*x^2+a+8*x)+1/16/(4+a)/(3+a)*sum((_R^2*(4+a)+2*(a+2 )*_R-a)/(-_R^3+3*_R^2-4*_R+2)*ln(x-_R),_R=RootOf(_Z^4-4*_Z^3+8*_Z^2-8*_Z-a ))
Timed out. \[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (185) = 370\).
Time = 18.36 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.49 \[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\frac {- a + x^{3} \left (- a - 4\right ) + x^{2} \left (a + 6\right ) + x \left (- a - 8\right )}{- 4 a^{3} - 28 a^{2} - 48 a + x^{4} \cdot \left (4 a^{2} + 28 a + 48\right ) + x^{3} \left (- 16 a^{2} - 112 a - 192\right ) + x^{2} \cdot \left (32 a^{2} + 224 a + 384\right ) + x \left (- 32 a^{2} - 224 a - 384\right )} + \operatorname {RootSum} {\left (t^{4} \cdot \left (65536 a^{9} + 2162688 a^{8} + 31653888 a^{7} + 269680640 a^{6} + 1473773568 a^{5} + 5357174784 a^{4} + 12952010752 a^{3} + 20082327552 a^{2} + 18119393280 a + 7247757312\right ) + t^{2} \left (- 9728 a^{6} - 209408 a^{5} - 1878016 a^{4} - 8986624 a^{3} - 24215552 a^{2} - 34865152 a - 20971520\right ) + t \left (256 a^{5} + 5888 a^{4} + 53248 a^{3} + 237568 a^{2} + 524288 a + 458752\right ) - a^{4} + 144 a^{3} + 1024 a^{2} + 1792 a, \left ( t \mapsto t \log {\left (x + \frac {4096 t^{3} a^{12} - 61440 t^{3} a^{11} - 5480448 t^{3} a^{10} - 111403008 t^{3} a^{9} - 1227173888 t^{3} a^{8} - 8682876928 t^{3} a^{7} - 42187440128 t^{3} a^{6} - 144630284288 t^{3} a^{5} - 350972280832 t^{3} a^{4} - 591750234112 t^{3} a^{3} - 660716126208 t^{3} a^{2} - 439848271872 t^{3} a - 132271570944 t^{3} - 28672 t^{2} a^{10} - 993280 t^{2} a^{9} - 15400960 t^{2} a^{8} - 140742656 t^{2} a^{7} - 839462912 t^{2} a^{6} - 3414427648 t^{2} a^{5} - 9590087680 t^{2} a^{4} - 18363547648 t^{2} a^{3} - 22938255360 t^{2} a^{2} - 16873684992 t^{2} a - 5549064192 t^{2} - 848 t a^{9} - 6096 t a^{8} + 174608 t a^{7} + 3323792 t a^{6} + 26276224 t a^{5} + 119009280 t a^{4} + 332017664 t a^{3} + 566497280 t a^{2} + 544112640 t a + 225837056 t + 11 a^{8} + 958 a^{7} + 17419 a^{6} + 142964 a^{5} + 632632 a^{4} + 1567552 a^{3} + 2049792 a^{2} + 1100800 a}{a^{8} + 870 a^{7} + 18289 a^{6} + 165176 a^{5} + 824560 a^{4} + 2452288 a^{3} + 4340224 a^{2} + 4229120 a + 1748992} \right )} \right )\right )} \]
(-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 + 5357 174784*a**4 + 12952010752*a**3 + 20082327552*a**2 + 18119393280*a + 724775 7312) + _t**2*(-9728*a**6 - 209408*a**5 - 1878016*a**4 - 8986624*a**3 - 24 215552*a**2 - 34865152*a - 20971520) + _t*(256*a**5 + 5888*a**4 + 53248*a* *3 + 237568*a**2 + 524288*a + 458752) - 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 - 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 - 43984827 1872*_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 - 3414 427648*_t**2*a**5 - 9590087680*_t**2*a**4 - 18363547648*_t**2*a**3 - 22938 255360*_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 + 1190 09280*_t*a**4 + 332017664*_t*a**3 + 566497280*_t*a**2 + 544112640*_t*a + 2 25837056*_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**...
\[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\int { \frac {x^{2}}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{2}} \,d x } \]
-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)
\[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\int { \frac {x^{2}}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{2}} \,d x } \]
Time = 9.44 (sec) , antiderivative size = 1218, normalized size of antiderivative = 5.41 \[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\text {Too large to display} \]
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 + 2162688* a^8*z^4 + 65536*a^9*z^4 + 18119393280*a*z^4 + 20082327552*a^2*z^4 + 147377 3568*a^5*z^4 + 5357174784*a^4*z^4 + 269680640*a^6*z^4 + 7247757312*z^4 - 2 4215552*a^2*z^2 - 8986624*a^3*z^2 - 1878016*a^4*z^2 - 209408*a^5*z^2 - 972 8*a^6*z^2 - 34865152*a*z^2 - 20971520*z^2 + 237568*a^2*z + 53248*a^3*z + 5 888*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*(81 6*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 + 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 + 2 37568*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 + 31 653888*a^7*z^4 + 2162688*a^8*z^4 + 65536*a^9*z^4 + 18119393280*a*z^4 + 200 82327552*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 + 23...