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

Optimal. Leaf size=349 \[ \frac{(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}-\frac{3 \left (7 a^2+\left (4 \sqrt{a+4}+47\right ) a+14 \sqrt{a+4}+80\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{64 (a+3)^2 (a+4)^{5/2} \sqrt{1-\sqrt{a+4}}}-\frac{3 \left (-\frac{7 a^2+47 a+80}{\sqrt{a+4}}+4 a+14\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{64 (a+3)^2 (a+4)^2 \sqrt{\sqrt{a+4}+1}}+\frac{3 \left ((x-1)^2+1\right )}{16 (a+4)^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac{(x-1)^2+1}{8 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}+\frac{(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{32 (a+3)^2 (a+4)^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac{3 \tanh ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a+4}}\right )}{16 (a+4)^{5/2}} \]

[Out]

(1 + (-1 + x)^2)/(8*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^2) + (3*(1 + (-1 + x)^2))/(16*(4 + a)^2*(3 + a
- 2*(-1 + x)^2 - (-1 + x)^4)) + ((5 + a + (-1 + x)^2)*(-1 + x))/(8*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (
-1 + x)^4)^2) + (((6 + a)*(25 + 7*a) + 6*(7 + 2*a)*(-1 + x)^2)*(-1 + x))/(32*(3 + a)^2*(4 + a)^2*(3 + a - 2*(-
1 + x)^2 - (-1 + x)^4)) - (3*(80 + 7*a^2 + 14*Sqrt[4 + a] + a*(47 + 4*Sqrt[4 + a]))*ArcTan[(-1 + x)/Sqrt[1 - S
qrt[4 + a]]])/(64*(3 + a)^2*(4 + a)^(5/2)*Sqrt[1 - Sqrt[4 + a]]) - (3*(14 + 4*a - (80 + 47*a + 7*a^2)/Sqrt[4 +
a])*ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]])/(64*(3 + a)^2*(4 + a)^2*Sqrt[1 + Sqrt[4 + a]]) + (3*ArcTanh[(1 +
(-1 + x)^2)/Sqrt[4 + a]])/(16*(4 + a)^(5/2))

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Rubi [A]  time = 0.368768, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {1680, 1673, 1092, 1178, 1166, 204, 1107, 614, 618, 206} \[ \frac{(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}-\frac{3 \left (7 a^2+\left (4 \sqrt{a+4}+47\right ) a+14 \sqrt{a+4}+80\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{64 (a+3)^2 (a+4)^{5/2} \sqrt{1-\sqrt{a+4}}}-\frac{3 \left (-\frac{7 a^2+47 a+80}{\sqrt{a+4}}+4 a+14\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{64 (a+3)^2 (a+4)^2 \sqrt{\sqrt{a+4}+1}}+\frac{3 \left ((x-1)^2+1\right )}{16 (a+4)^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac{(x-1)^2+1}{8 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}+\frac{(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{32 (a+3)^2 (a+4)^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac{3 \tanh ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a+4}}\right )}{16 (a+4)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + (-1 + x)^2)/(8*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^2) + (3*(1 + (-1 + x)^2))/(16*(4 + a)^2*(3 + a
- 2*(-1 + x)^2 - (-1 + x)^4)) + ((5 + a + (-1 + x)^2)*(-1 + x))/(8*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (
-1 + x)^4)^2) + (((6 + a)*(25 + 7*a) + 6*(7 + 2*a)*(-1 + x)^2)*(-1 + x))/(32*(3 + a)^2*(4 + a)^2*(3 + a - 2*(-
1 + x)^2 - (-1 + x)^4)) - (3*(80 + 7*a^2 + 14*Sqrt[4 + a] + a*(47 + 4*Sqrt[4 + a]))*ArcTan[(-1 + x)/Sqrt[1 - S
qrt[4 + a]]])/(64*(3 + a)^2*(4 + a)^(5/2)*Sqrt[1 - Sqrt[4 + a]]) - (3*(14 + 4*a - (80 + 47*a + 7*a^2)/Sqrt[4 +
a])*ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]])/(64*(3 + a)^2*(4 + a)^2*Sqrt[1 + Sqrt[4 + a]]) + (3*ArcTanh[(1 +
(-1 + x)^2)/Sqrt[4 + a]])/(16*(4 + a)^(5/2))

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]

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 1092

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

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 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 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 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 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 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])

Rubi steps

\begin{align*} \int \frac{x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx &=\operatorname{Subst}\left (\int \frac{1+x}{\left (3+a-2 x^2-x^4\right )^3} \, dx,x,-1+x\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{\left (3+a-2 x^2-x^4\right )^3} \, dx,x,-1+x\right )+\operatorname{Subst}\left (\int \frac{x}{\left (3+a-2 x^2-x^4\right )^3} \, dx,x,-1+x\right )\\ &=\frac{\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (3+a-2 x-x^2\right )^3} \, dx,x,(-1+x)^2\right )-\frac{\operatorname{Subst}\left (\int \frac{4+2 (3+a)-4 (4+4 (3+a))-10 x^2}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )}{16 \left (12+7 a+a^2\right )}\\ &=\frac{1+(-1+x)^2}{8 (4+a) \left (3+a-2 (1-x)^2-(1-x)^4\right )^2}-\frac{\left ((6+a) (25+7 a)+6 (7+2 a) (1-x)^2\right ) (1-x)}{32 \left (12+7 a+a^2\right )^2 \left (3+a-2 (1-x)^2-(1-x)^4\right )}+\frac{\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\left (3+a-2 x-x^2\right )^2} \, dx,x,(-1+x)^2\right )}{8 (4+a)}+\frac{\operatorname{Subst}\left (\int \frac{12 \left (94+51 a+7 a^2\right )+24 (7+2 a) x^2}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )}{128 \left (12+7 a+a^2\right )^2}\\ &=\frac{1+(-1+x)^2}{8 (4+a) \left (3+a-2 (1-x)^2-(1-x)^4\right )^2}+\frac{3 \left (1+(-1+x)^2\right )}{16 (4+a)^2 \left (3+a-2 (1-x)^2-(1-x)^4\right )}-\frac{\left ((6+a) (25+7 a)+6 (7+2 a) (1-x)^2\right ) (1-x)}{32 \left (12+7 a+a^2\right )^2 \left (3+a-2 (1-x)^2-(1-x)^4\right )}+\frac{\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{3+a-2 x-x^2} \, dx,x,(-1+x)^2\right )}{16 (4+a)^2}+\frac{\left (3 \left (14+4 a-\frac{80+47 a+7 a^2}{\sqrt{4+a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-\sqrt{4+a}-x^2} \, dx,x,-1+x\right )}{64 \left (12+7 a+a^2\right )^2}+\frac{\left (3 \left (80+7 a^2+14 \sqrt{4+a}+a \left (47+4 \sqrt{4+a}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\sqrt{4+a}-x^2} \, dx,x,-1+x\right )}{64 \sqrt{4+a} \left (12+7 a+a^2\right )^2}\\ &=\frac{1+(-1+x)^2}{8 (4+a) \left (3+a-2 (1-x)^2-(1-x)^4\right )^2}+\frac{3 \left (1+(-1+x)^2\right )}{16 (4+a)^2 \left (3+a-2 (1-x)^2-(1-x)^4\right )}-\frac{\left ((6+a) (25+7 a)+6 (7+2 a) (1-x)^2\right ) (1-x)}{32 \left (12+7 a+a^2\right )^2 \left (3+a-2 (1-x)^2-(1-x)^4\right )}+\frac{\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac{3 \left (80+47 a+7 a^2+\sqrt{4+a} (14+4 a)\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{4+a}}}\right )}{64 (3+a)^2 (4+a)^{5/2} \sqrt{1-\sqrt{4+a}}}+\frac{3 \left (14+4 a-\frac{80+47 a+7 a^2}{\sqrt{4+a}}\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )}{64 \left (12+7 a+a^2\right )^2 \sqrt{1+\sqrt{4+a}}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{4 (4+a)-x^2} \, dx,x,-2 \left (1+(-1+x)^2\right )\right )}{8 (4+a)^2}\\ &=\frac{1+(-1+x)^2}{8 (4+a) \left (3+a-2 (1-x)^2-(1-x)^4\right )^2}+\frac{3 \left (1+(-1+x)^2\right )}{16 (4+a)^2 \left (3+a-2 (1-x)^2-(1-x)^4\right )}-\frac{\left ((6+a) (25+7 a)+6 (7+2 a) (1-x)^2\right ) (1-x)}{32 \left (12+7 a+a^2\right )^2 \left (3+a-2 (1-x)^2-(1-x)^4\right )}+\frac{\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac{3 \left (80+47 a+7 a^2+\sqrt{4+a} (14+4 a)\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{4+a}}}\right )}{64 (3+a)^2 (4+a)^{5/2} \sqrt{1-\sqrt{4+a}}}+\frac{3 \left (14+4 a-\frac{80+47 a+7 a^2}{\sqrt{4+a}}\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )}{64 \left (12+7 a+a^2\right )^2 \sqrt{1+\sqrt{4+a}}}+\frac{3 \tanh ^{-1}\left (\frac{1+(-1+x)^2}{\sqrt{4+a}}\right )}{16 (4+a)^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.119132, size = 284, normalized size = 0.81 \[ \frac{1}{128} \left (-\frac{3 \text{RootSum}\left [-\text{$\#$1}^4+4 \text{$\#$1}^3-8 \text{$\#$1}^2+8 \text{$\#$1}+a\& ,\frac{4 \text{$\#$1}^2 a \log (x-\text{$\#$1})+14 \text{$\#$1}^2 \log (x-\text{$\#$1})+3 a^2 \log (x-\text{$\#$1})+4 \text{$\#$1} a^2 \log (x-\text{$\#$1})+31 a \log (x-\text{$\#$1})+16 \text{$\#$1} a \log (x-\text{$\#$1})+72 \log (x-\text{$\#$1})+8 \text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^3-3 \text{$\#$1}^2+4 \text{$\#$1}-2}\& \right ]}{\left (a^2+7 a+12\right )^2}+\frac{4 \left (a^2 \left (6 x^2-5 x+5\right )+a \left (12 x^3+31 x-7\right )+6 \left (7 x^3-12 x^2+28 x-14\right )\right )}{(a+3)^2 (a+4)^2 \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )}+\frac{16 \left (a x^2-a x+a+x^3+2 x\right )}{(a+3) (a+4) \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((16*(a + 2*x - a*x + a*x^2 + x^3))/((3 + a)*(4 + a)*(a - x*(-8 + 8*x - 4*x^2 + x^3))^2) + (4*(a^2*(5 - 5*x +
6*x^2) + 6*(-14 + 28*x - 12*x^2 + 7*x^3) + a*(-7 + 31*x + 12*x^3)))/((3 + a)^2*(4 + a)^2*(a - x*(-8 + 8*x - 4*
x^2 + x^3))) - (3*RootSum[a + 8*#1 - 8*#1^2 + 4*#1^3 - #1^4 & , (72*Log[x - #1] + 31*a*Log[x - #1] + 3*a^2*Log
[x - #1] + 8*Log[x - #1]*#1 + 16*a*Log[x - #1]*#1 + 4*a^2*Log[x - #1]*#1 + 14*Log[x - #1]*#1^2 + 4*a*Log[x - #
1]*#1^2)/(-2 + 4*#1 - 3*#1^2 + #1^3) & ])/(12 + 7*a + a^2)^2)/128

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Maple [C]  time = 0.016, size = 405, normalized size = 1.2 \begin{align*} -{\frac{1}{ \left ({x}^{4}-4\,{x}^{3}+8\,{x}^{2}-a-8\,x \right ) ^{2}} \left ({\frac{ \left ( 6\,a+21 \right ){x}^{7}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}+{\frac{ \left ( 3\,{a}^{2}-24\,a-120 \right ){x}^{6}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}-{\frac{ \left ( 29\,{a}^{2}-127\,a-792 \right ){x}^{5}}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}}+{\frac{ \left ( 73\,{a}^{2}-227\,a-1668 \right ){x}^{4}}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}}-{\frac{ \left ( 62\,{a}^{2}-103\,a-1104 \right ){x}^{3}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}-{\frac{ \left ( 5\,{a}^{3}-26\,{a}^{2}+140\,a+1008 \right ){x}^{2}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}+{\frac{ \left ( 9\,{a}^{3}-51\,{a}^{2}-120\,a+576 \right ) x}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}}-{\frac{3\,a \left ( 3\,{a}^{2}+7\,a-12 \right ) }{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}} \right ) }-{\frac{3}{128\,{a}^{4}+1792\,{a}^{3}+9344\,{a}^{2}+21504\,a+18432}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}-8\,{\it \_Z}-a \right ) }{\frac{ \left ( 72+2\, \left ( 7+2\,a \right ){{\it \_R}}^{2}+4\, \left ({a}^{2}+4\,a+2 \right ){\it \_R}+3\,{a}^{2}+31\,a \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+4\,{\it \_R}-2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3/16*(7+2*a)/(a^4+14*a^3+73*a^2+168*a+144)*x^7+3/16*(a^2-8*a-40)/(a^4+14*a^3+73*a^2+168*a+144)*x^6-1/32*(29*
a^2-127*a-792)/(a^4+14*a^3+73*a^2+168*a+144)*x^5+1/32*(73*a^2-227*a-1668)/(a^4+14*a^3+73*a^2+168*a+144)*x^4-1/
16*(62*a^2-103*a-1104)/(a^4+14*a^3+73*a^2+168*a+144)*x^3-1/16*(5*a^3-26*a^2+140*a+1008)/(a^4+14*a^3+73*a^2+168
*a+144)*x^2+3/32*(3*a^3-17*a^2-40*a+192)/(a^4+14*a^3+73*a^2+168*a+144)*x-3/32*a*(3*a^2+7*a-12)/(a^4+14*a^3+73*
a^2+168*a+144))/(x^4-4*x^3+8*x^2-a-8*x)^2-3/128/(a^4+14*a^3+73*a^2+168*a+144)*sum((72+2*(7+2*a)*_R^2+4*(a^2+4*
a+2)*_R+3*a^2+31*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))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError