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

Optimal. Leaf size=474 \[ \frac{(x-1) \left (a+(x-1)^2+5\right )}{2 \left (a^2+7 a+12\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1)^2+1}{2 (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{\left (1-\sqrt{a+4}\right ) (x-1) \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right )}{2 (a+3) (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{2 (a+4) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{\left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{2 (a+3) (a+4) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]

[Out]

(1 + (-1 + x)^2)/(2*(4 + a)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + ((5 + a + (-1 + x)^2)*(-1 + x))/(2*(12
+ 7*a + a^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) - ((1 - Sqrt[4 + a])*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*
(-1 + x))/(2*(3 + a)*(4 + a)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + ((1 - Sqrt[4 + a])*Sqrt[1 + Sqrt[4 + a
]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*EllipticE[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 -
Sqrt[4 + a])])/(2*(3 + a)*(4 + a)*Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*
Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + (Sqrt[1 + Sqrt[4 + a]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*EllipticF
[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(2*(4 + a)*Sqrt[(1 + (-1 + x)^2/
(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])

________________________________________________________________________________________

Rubi [A]  time = 0.43514, antiderivative size = 474, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1680, 1673, 1092, 1202, 531, 418, 492, 411, 1107, 613} \[ \frac{(x-1) \left (a+(x-1)^2+5\right )}{2 \left (a^2+7 a+12\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1)^2+1}{2 (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{\left (1-\sqrt{a+4}\right ) (x-1) \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right )}{2 (a+3) (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{2 (a+4) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{\left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{2 (a+3) (a+4) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + (-1 + x)^2)/(2*(4 + a)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + ((5 + a + (-1 + x)^2)*(-1 + x))/(2*(12
+ 7*a + a^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) - ((1 - Sqrt[4 + a])*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*
(-1 + x))/(2*(3 + a)*(4 + a)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + ((1 - Sqrt[4 + a])*Sqrt[1 + Sqrt[4 + a
]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*EllipticE[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 -
Sqrt[4 + a])])/(2*(3 + a)*(4 + a)*Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*
Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + (Sqrt[1 + Sqrt[4 + a]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*EllipticF
[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(2*(4 + a)*Sqrt[(1 + (-1 + x)^2/
(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])

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 1202

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

Rule 531

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

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 613

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

Rubi steps

\begin{align*} \int \frac{x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx &=\operatorname{Subst}\left (\int \frac{1+x}{\left (3+a-2 x^2-x^4\right )^{3/2}} \, dx,x,-1+x\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{\left (3+a-2 x^2-x^4\right )^{3/2}} \, dx,x,-1+x\right )+\operatorname{Subst}\left (\int \frac{x}{\left (3+a-2 x^2-x^4\right )^{3/2}} \, dx,x,-1+x\right )\\ &=\frac{\left (5+a+(-1+x)^2\right ) (-1+x)}{2 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (3+a-2 x-x^2\right )^{3/2}} \, dx,x,(-1+x)^2\right )-\frac{\operatorname{Subst}\left (\int \frac{-2 (3+a)+2 x^2}{\sqrt{3+a-2 x^2-x^4}} \, dx,x,-1+x\right )}{4 \left (12+7 a+a^2\right )}\\ &=\frac{1+(-1+x)^2}{2 (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4}}+\frac{\left (5+a+(-1+x)^2\right ) (-1+x)}{2 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}-\frac{\left (\sqrt{1-\frac{2 (-1+x)^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 (-1+x)^2}{-2+2 \sqrt{4+a}}}\right ) \operatorname{Subst}\left (\int \frac{-2 (3+a)+2 x^2}{\sqrt{1-\frac{2 x^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 x^2}{-2+2 \sqrt{4+a}}}} \, dx,x,-1+x\right )}{4 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac{1+(-1+x)^2}{2 (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4}}+\frac{\left (5+a+(-1+x)^2\right ) (-1+x)}{2 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}-\frac{\left (\sqrt{1-\frac{2 (-1+x)^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 (-1+x)^2}{-2+2 \sqrt{4+a}}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{2 x^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 x^2}{-2+2 \sqrt{4+a}}}} \, dx,x,-1+x\right )}{2 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}+\frac{\left ((3+a) \sqrt{1-\frac{2 (-1+x)^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 (-1+x)^2}{-2+2 \sqrt{4+a}}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{2 x^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 x^2}{-2+2 \sqrt{4+a}}}} \, dx,x,-1+x\right )}{2 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac{1+(-1+x)^2}{2 (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4}}+\frac{\left (1-\sqrt{4+a}\right ) \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) (1-x)}{2 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4}}+\frac{\left (5+a+(-1+x)^2\right ) (-1+x)}{2 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}-\frac{\sqrt{1+\sqrt{4+a}} \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )|-\frac{2 \sqrt{4+a}}{1-\sqrt{4+a}}\right )}{2 (4+a) \sqrt{\frac{1+\frac{(1-x)^2}{1-\sqrt{4+a}}}{1+\frac{(1-x)^2}{1+\sqrt{4+a}}}} \sqrt{3+a-2 (1-x)^2-(1-x)^4}}+\frac{\left (\left (1-\sqrt{4+a}\right ) \sqrt{1-\frac{2 (-1+x)^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 (-1+x)^2}{-2+2 \sqrt{4+a}}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{2 x^2}{-2+2 \sqrt{4+a}}}}{\left (1-\frac{2 x^2}{-2-2 \sqrt{4+a}}\right )^{3/2}} \, dx,x,-1+x\right )}{2 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac{1+(-1+x)^2}{2 (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4}}+\frac{\left (1-\sqrt{4+a}\right ) \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) (1-x)}{2 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4}}+\frac{\left (5+a+(-1+x)^2\right ) (-1+x)}{2 \left (12+7 a+a^2\right ) \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}-\frac{\left (1-\sqrt{4+a}\right ) \sqrt{1+\sqrt{4+a}} \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) E\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )|-\frac{2 \sqrt{4+a}}{1-\sqrt{4+a}}\right )}{2 \left (12+7 a+a^2\right ) \sqrt{\frac{1+\frac{(1-x)^2}{1-\sqrt{4+a}}}{1+\frac{(1-x)^2}{1+\sqrt{4+a}}}} \sqrt{3+a-2 (1-x)^2-(1-x)^4}}-\frac{\sqrt{1+\sqrt{4+a}} \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )|-\frac{2 \sqrt{4+a}}{1-\sqrt{4+a}}\right )}{2 (4+a) \sqrt{\frac{1+\frac{(1-x)^2}{1-\sqrt{4+a}}}{1+\frac{(1-x)^2}{1+\sqrt{4+a}}}} \sqrt{3+a-2 (1-x)^2-(1-x)^4}}\\ \end{align*}

Mathematica [B]  time = 6.07987, size = 3593, normalized size = 7.58 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-a - 2*x + a*x - a*x^2 - x^3)*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2)/(2*(3 + a)*(4 + a)*(-a - 8*x + 8*x^2 - 4*x^
3 + x^4)*(a - x*(-8 + 8*x - 4*x^2 + x^3))^(3/2)) + ((a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2)*((4*(-Sqrt[-1 - Sqrt
[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-
1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 -
Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt
[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt
[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))
]*EllipticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))
/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4
+ a]] - Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] -
Sqrt[-1 + Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-
1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (2*a*(-Sqrt[-1 - Sqrt[4 + a]
] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqr
t[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1
- Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]
] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + S
qrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Ellip
ticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt
[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4 + a]] -
Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1
+ Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqr
t[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (4*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[
-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])
*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 +
a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1
+ Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]]
+ x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*((-1 - Sqrt[-1 -
Sqrt[4 + a]])*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4
+ a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1
- Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2] + 2*Sqrt[-1 -
Sqrt[4 + a]]*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 +
Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] +
x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4
+ a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2]))/(Sqrt[-1 - Sqrt[4 + a
]]*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) - ((-1 + Sqrt[-1 - S
qrt[4 + a]] + x)*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x)*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x) + 2*(Sqrt[-1 - Sqrt[4 + a
]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqr
t[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1
- Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]]
+ Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sq
rt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*(((Sqr
t[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*EllipticE[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[
4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 -
Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 +
Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]) + ((-((-1 - Sqrt[-1 - Sqrt[4 + a]])*(-2 - Sqrt[-1 - Sqrt[4 + a]]
- Sqrt[-1 + Sqrt[4 + a]])) + (-1 + Sqrt[-1 - Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]]))*
EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((
Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]]
+ Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]*(-
Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])) + (4*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 +
a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqr
t[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1
- Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1
+ Sqrt[4 + a]])^2])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])))/Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]))
/(2*(3 + a)*(4 + a)*(a - x*(-8 + 8*x - 4*x^2 + x^3))^(3/2))

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Maple [B]  time = 0.016, size = 2616, normalized size = 5.5 \begin{align*} \text{result too large to display} \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)^(3/2),x)

[Out]

2*(1/4/(a^2+7*a+12)*x^3+1/4*a/(a^2+7*a+12)*x^2-1/4*(a-2)/(a^2+7*a+12)*x+1/4*a/(a^2+7*a+12))/(-x^4+4*x^3-8*x^2+
a+8*x)^(1/2)-(2/(a^2+7*a+12)+1/2*(a-2)/(a^2+7*a+12))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*((-(-1-(4
+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2)
)^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(x-1+(-1+(4+a)^(1/2))^(1/2))^2*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1-(-
1-(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(-2*
(-1+(4+a)^(1/2))^(1/2)*(x-1+(-1-(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+
(4+a)^(1/2))^(1/2)))^(1/2)/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-1+(4+a)^(1/2))^(1/2)/(-(x-1-(-1+
(4+a)^(1/2))^(1/2))*(x-1+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1-(4+a)^(1/2))^(1/2))*(x-1+(-1-(4+a)^(1/2))^(1/2)))^(1
/2)*EllipticF(((-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2)
)^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))
^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-
(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^(1/2))-((1+a)/(a^2+7*a+12)-a/(a^2+7*a+12))*((-1-(4+a)^(1/2))^(1/2)
+(-1+(4+a)^(1/2))^(1/2))*((-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-
(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(x-1+(-1+(4+a)^(1/2))^(1/2))^2*
(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1-(-1-(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(
-1+(4+a)^(1/2))^(1/2)))^(1/2)*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1+(-1-(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)
-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/
(-1+(4+a)^(1/2))^(1/2)/(-(x-1-(-1+(4+a)^(1/2))^(1/2))*(x-1+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1-(4+a)^(1/2))^(1/2)
)*(x-1+(-1-(4+a)^(1/2))^(1/2)))^(1/2)*((1-(-1+(4+a)^(1/2))^(1/2))*EllipticF(((-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a
)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(
1/2))^(1/2)))^(1/2),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))
^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^(1/2
))+2*(-1+(4+a)^(1/2))^(1/2)*EllipticPi(((-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))
^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),(-(-1-(4+a)^(1/2)
)^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2)),((-(-1-(4+a)^(1/2))^(1/2)-(-1
+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))
^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^(1/2)))-1/2/(a^2+7*a+12)*((x-1-(-1+(4+a)^(1/2))^(1/2)
)*(x-1-(-1-(4+a)^(1/2))^(1/2))*(x-1+(-1-(4+a)^(1/2))^(1/2))+((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*((
-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a
)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(x-1+(-1+(4+a)^(1/2))^(1/2))^2*(-2*(-1+(4+a)^(1/2))^(1/2)*
(x-1-(-1-(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/
2)*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1+(-1-(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x
-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(-1/2*((1-(-1+(4+a)^(1/2))^(1/2))*(1+(-1+(4+a)^(1/2))^(1/2))-(1-(-1-(4+a)^(1
/2))^(1/2))*(1+(-1+(4+a)^(1/2))^(1/2))+(1-(-1-(4+a)^(1/2))^(1/2))*(1-(-1+(4+a)^(1/2))^(1/2))+(1-(-1+(4+a)^(1/2
))^(1/2))^2)/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-1+(4+a)^(1/2))^(1/2)*EllipticF(((-(-1-(4+a)^(1
/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2
))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/
2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^
(1/2))^(1/2)))^(1/2))-1/2*(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*EllipticE(((-(-1-(4+a)^(1/2))^(1/2)
+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-
1+(4+a)^(1/2))^(1/2)))^(1/2),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+
a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/
2)))^(1/2))/(-1+(4+a)^(1/2))^(1/2)-4/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*EllipticPi(((-(-1-(4+a)^
(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1
/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/
2)-(-1+(4+a)^(1/2))^(1/2)),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)
^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)
))^(1/2))))/(-(x-1-(-1+(4+a)^(1/2))^(1/2))*(x-1+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1-(4+a)^(1/2))^(1/2))*(x-1+(-1-
(4+a)^(1/2))^(1/2)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac{3}{2}}}\,{d x} \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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x}{x^{8} - 8 \, x^{7} + 32 \, x^{6} - 2 \,{\left (a - 64\right )} x^{4} - 80 \, x^{5} + 8 \,{\left (a - 16\right )} x^{3} - 16 \,{\left (a - 4\right )} x^{2} + a^{2} + 16 \, a x}, x\right ) \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)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)*x/(x^8 - 8*x^7 + 32*x^6 - 2*(a - 64)*x^4 - 80*x^5 + 8*(a - 16)*x
^3 - 16*(a - 4)*x^2 + a^2 + 16*a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac{3}{2}}}\, dx \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)**(3/2),x)

[Out]

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

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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)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError