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

Optimal. Leaf size=558 \[ \frac{3}{16} (a+4) \left ((x-1)^2+1\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{1}{8} \left ((x-1)^2+1\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{2}{35} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}-\frac{16 (2 a+7) \left (1-\sqrt{a+4}\right ) (x-1) \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right )}{35 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{3}{16} (a+4)^2 \tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac{4 (a+3) (5 a+16) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{35 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{16 (2 a+7) \left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{35 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]

[Out]

(3*(4 + a)*(1 + (-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])/16 + ((1 + (-1 + x)^2)*(3 + a - 2*(-1 + x
)^2 - (-1 + x)^4)^(3/2))/8 - (16*(7 + 2*a)*(1 - Sqrt[4 + a])*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*(-1 + x))/(35*
Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + (2*(13 + 5*a - 3*(-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4
]*(-1 + x))/35 + ((3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x))/7 + (3*(4 + a)^2*ArcTan[(1 + (-1 + x)^2)
/Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]])/16 + (16*(7 + 2*a)*(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]
)])/(35*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]) + (4*(3 + a)*(16 + 5*a)*Sqrt[1 + Sqrt[4 + a]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*EllipticF[ArcT
an[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(35*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])

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Rubi [A]  time = 0.565067, antiderivative size = 558, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1680, 1673, 1091, 1176, 1202, 531, 418, 492, 411, 1107, 612, 621, 204} \[ \frac{3}{16} (a+4) \left ((x-1)^2+1\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{1}{8} \left ((x-1)^2+1\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{2}{35} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}-\frac{16 (2 a+7) \left (1-\sqrt{a+4}\right ) (x-1) \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right )}{35 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{3}{16} (a+4)^2 \tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac{4 (a+3) (5 a+16) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{35 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{16 (2 a+7) \left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{35 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(4 + a)*(1 + (-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])/16 + ((1 + (-1 + x)^2)*(3 + a - 2*(-1 + x
)^2 - (-1 + x)^4)^(3/2))/8 - (16*(7 + 2*a)*(1 - Sqrt[4 + a])*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*(-1 + x))/(35*
Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + (2*(13 + 5*a - 3*(-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4
]*(-1 + x))/35 + ((3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x))/7 + (3*(4 + a)^2*ArcTan[(1 + (-1 + x)^2)
/Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]])/16 + (16*(7 + 2*a)*(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]
)])/(35*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]) + (4*(3 + a)*(16 + 5*a)*Sqrt[1 + Sqrt[4 + a]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*EllipticF[ArcT
an[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(35*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 1091

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a + b*x^2 + c*x^4)^p)/(4*p + 1), x] + Dis
t[(2*p)/(4*p + 1), Int[(2*a + b*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] && GtQ[p, 0] && IntegerQ[2*p]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(2*b*e*p + c*d*(4*p
+ 3) + c*e*(4*p + 1)*x^2)*(a + b*x^2 + c*x^4)^p)/(c*(4*p + 1)*(4*p + 3)), x] + Dist[(2*p)/(c*(4*p + 1)*(4*p +
3)), Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*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] && GtQ[p, 0] && FractionQ[p] && 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 612

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

Rule 621

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

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])

Rubi steps

\begin{align*} \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int (1+x) \left (3+a-2 x^2-x^4\right )^{3/2} \, dx,x,-1+x\right )\\ &=\operatorname{Subst}\left (\int \left (3+a-2 x^2-x^4\right )^{3/2} \, dx,x,-1+x\right )+\operatorname{Subst}\left (\int x \left (3+a-2 x^2-x^4\right )^{3/2} \, dx,x,-1+x\right )\\ &=\frac{1}{7} \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac{3}{7} \operatorname{Subst}\left (\int \left (2 (3+a)-2 x^2\right ) \sqrt{3+a-2 x^2-x^4} \, dx,x,-1+x\right )+\frac{1}{2} \operatorname{Subst}\left (\int \left (3+a-2 x-x^2\right )^{3/2} \, dx,x,(-1+x)^2\right )\\ &=\frac{1}{8} \left (3+a-2 (1-x)^2-(1-x)^4\right )^{3/2} \left (1+(-1+x)^2\right )-\frac{2}{35} \left (13+5 a-3 (1-x)^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{7} \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)-\frac{1}{35} \operatorname{Subst}\left (\int \frac{-4 (3+a) (16+5 a)+16 (7+2 a) x^2}{\sqrt{3+a-2 x^2-x^4}} \, dx,x,-1+x\right )+\frac{1}{8} (3 (4+a)) \operatorname{Subst}\left (\int \sqrt{3+a-2 x-x^2} \, dx,x,(-1+x)^2\right )\\ &=\frac{3}{16} (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4} \left (1+(-1+x)^2\right )+\frac{1}{8} \left (3+a-2 (1-x)^2-(1-x)^4\right )^{3/2} \left (1+(-1+x)^2\right )-\frac{2}{35} \left (13+5 a-3 (1-x)^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{7} \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac{1}{16} \left (3 (4+a)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+a-2 x-x^2}} \, dx,x,(-1+x)^2\right )-\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{-4 (3+a) (16+5 a)+16 (7+2 a) 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 )}{35 \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac{3}{16} (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4} \left (1+(-1+x)^2\right )+\frac{1}{8} \left (3+a-2 (1-x)^2-(1-x)^4\right )^{3/2} \left (1+(-1+x)^2\right )-\frac{2}{35} \left (13+5 a-3 (1-x)^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{7} \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac{1}{8} \left (3 (4+a)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,-\frac{2 \left (1+(-1+x)^2\right )}{\sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\right )-\frac{\left (16 (7+2 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{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 )}{35 \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}+\frac{\left (4 (3+a) (16+5 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 )}{35 \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac{3}{16} (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4} \left (1+(-1+x)^2\right )+\frac{1}{8} \left (3+a-2 (1-x)^2-(1-x)^4\right )^{3/2} \left (1+(-1+x)^2\right )+\frac{16 (7+2 a) \left (1-\sqrt{4+a}\right ) \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) (1-x)}{35 \sqrt{3+a-2 (1-x)^2-(1-x)^4}}-\frac{2}{35} \left (13+5 a-3 (1-x)^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{7} \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac{3}{16} (4+a)^2 \tan ^{-1}\left (\frac{1+(-1+x)^2}{\sqrt{3+a-2 (1-x)^2-(1-x)^4}}\right )-\frac{4 (3+a) (16+5 a) \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 )}{35 \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 (16 (7+2 a) \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 )}{35 \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac{3}{16} (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4} \left (1+(-1+x)^2\right )+\frac{1}{8} \left (3+a-2 (1-x)^2-(1-x)^4\right )^{3/2} \left (1+(-1+x)^2\right )+\frac{16 (7+2 a) \left (1-\sqrt{4+a}\right ) \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) (1-x)}{35 \sqrt{3+a-2 (1-x)^2-(1-x)^4}}-\frac{2}{35} \left (13+5 a-3 (1-x)^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{7} \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac{3}{16} (4+a)^2 \tan ^{-1}\left (\frac{1+(-1+x)^2}{\sqrt{3+a-2 (1-x)^2-(1-x)^4}}\right )-\frac{16 (7+2 a) \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 )}{35 \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{4 (3+a) (16+5 a) \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 )}{35 \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 = 6.12917, size = 7235, normalized size = 12.97 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.018, size = 2694, normalized size = 4.8 \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]

-1/8*x^6*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+17/28*x^5*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-43/28*x^4*(-x^4+4*x^3-8*x^2+a
+8*x)^(1/2)+74/35*x^3*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(5/16*a-9/20)*x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(-11/56*
a-29/70)*x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(11/56*a+13/14)*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-(-(-11/56*a-29/70)*a-
11/14*a-26/7)*((-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))-(a^2-2*(5/16*a-9/20)*a+55/14*a+62/5)*((-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)))+(-32/35*a-16/5)*((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{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac{3}{2}} x\,{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^4 + 4*x^3 - 8*x^2 + a + 8*x)^(3/2)*x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (x^{5} - 4 \, x^{4} + 8 \, x^{3} - a x - 8 \, x^{2}\right )} \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, 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(-(x^5 - 4*x^4 + 8*x^3 - a*x - 8*x^2)*sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x), x)

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

[Out]

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