3.791 \(\int x^2 (a+8 x-8 x^2+4 x^3-x^4)^{3/2} \, dx\)
Optimal. Leaf size=585 \[ \frac{4 \left (21 a^2+111 a+140\right ) \left (1-\sqrt{a+4}\right ) (x-1) \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right )}{315 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{4 \left (21 a^2+111 a+140\right ) \left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{315 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{3}{8} (a+4) \left ((x-1)^2+1\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{1}{4} \left ((x-1)^2+1\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{2}{315} (x-1) \left (3 (7 a+20) (x-1)^2+2 (27 a+80)\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{3}{8} (a+4)^2 \tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac{4 (a+3) (33 a+100) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{315 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]
[Out]
(3*(4 + a)*(1 + (-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])/8 + ((1 + (-1 + x)^2)*(3 + a - 2*(-1 + x)
^2 - (-1 + x)^4)^(3/2))/4 + (4*(140 + 111*a + 21*a^2)*(1 - Sqrt[4 + a])*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*(-1
+ x))/(315*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + (2*(2*(80 + 27*a) + 3*(20 + 7*a)*(-1 + x)^2)*Sqrt[3 + a
- 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/315 + ((15 + 7*(-1 + x)^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(
-1 + x))/63 + (3*(4 + a)^2*ArcTan[(1 + (-1 + x)^2)/Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]])/8 - (4*(140 + 111
*a + 21*a^2)*(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])])/(315*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)*(100 + 33*a)*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])])/(315*Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*Sq
rt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])
________________________________________________________________________________________
Rubi [A] time = 0.678902, antiderivative size = 585, normalized size of antiderivative = 1.,
number of steps used = 15, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464,
Rules used = {1680, 1673, 1176, 1202, 531, 418, 492, 411, 12, 1107, 612, 621, 204}
\[ \frac{4 \left (21 a^2+111 a+140\right ) \left (1-\sqrt{a+4}\right ) (x-1) \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right )}{315 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{4 \left (21 a^2+111 a+140\right ) \left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{315 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{3}{8} (a+4) \left ((x-1)^2+1\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{1}{4} \left ((x-1)^2+1\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{2}{315} (x-1) \left (3 (7 a+20) (x-1)^2+2 (27 a+80)\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{3}{8} (a+4)^2 \tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac{4 (a+3) (33 a+100) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{315 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]
Antiderivative was successfully verified.
[In]
Int[x^2*(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])/8 + ((1 + (-1 + x)^2)*(3 + a - 2*(-1 + x)
^2 - (-1 + x)^4)^(3/2))/4 + (4*(140 + 111*a + 21*a^2)*(1 - Sqrt[4 + a])*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*(-1
+ x))/(315*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + (2*(2*(80 + 27*a) + 3*(20 + 7*a)*(-1 + x)^2)*Sqrt[3 + a
- 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/315 + ((15 + 7*(-1 + x)^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(
-1 + x))/63 + (3*(4 + a)^2*ArcTan[(1 + (-1 + x)^2)/Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]])/8 - (4*(140 + 111
*a + 21*a^2)*(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])])/(315*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)*(100 + 33*a)*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])])/(315*Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*Sq
rt[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 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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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^2 \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int (1+x)^2 \left (3+a-2 x^2-x^4\right )^{3/2} \, dx,x,-1+x\right )\\ &=\operatorname{Subst}\left (\int 2 x \left (3+a-2 x^2-x^4\right )^{3/2} \, dx,x,-1+x\right )+\operatorname{Subst}\left (\int \left (1+x^2\right ) \left (3+a-2 x^2-x^4\right )^{3/2} \, dx,x,-1+x\right )\\ &=\frac{1}{63} \left (15+7 (-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)-\frac{1}{21} \operatorname{Subst}\left (\int \left (-16 (3+a)-2 (20+7 a) x^2\right ) \sqrt{3+a-2 x^2-x^4} \, dx,x,-1+x\right )+2 \operatorname{Subst}\left (\int x \left (3+a-2 x^2-x^4\right )^{3/2} \, dx,x,-1+x\right )\\ &=-\frac{2}{315} \left (2 (80+27 a)+3 (20+7 a) (1-x)^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{63} \left (15+7 (-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac{1}{315} \operatorname{Subst}\left (\int \frac{4 (3+a) (100+33 a)+4 \left (140+111 a+21 a^2\right ) x^2}{\sqrt{3+a-2 x^2-x^4}} \, dx,x,-1+x\right )+\operatorname{Subst}\left (\int \left (3+a-2 x-x^2\right )^{3/2} \, dx,x,(-1+x)^2\right )\\ &=\frac{1}{4} \left (1+(-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}-\frac{2}{315} \left (2 (80+27 a)+3 (20+7 a) (1-x)^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{63} \left (15+7 (-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac{1}{4} (3 (4+a)) \operatorname{Subst}\left (\int \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) (100+33 a)+4 \left (140+111 a+21 a^2\right ) 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 )}{315 \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac{3}{8} (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4} \left (1+(-1+x)^2\right )+\frac{1}{4} \left (1+(-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}-\frac{2}{315} \left (2 (80+27 a)+3 (20+7 a) (1-x)^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{63} \left (15+7 (-1+x)^2\right ) \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}{\sqrt{3+a-2 x-x^2}} \, dx,x,(-1+x)^2\right )+\frac{\left (4 (3+a) (100+33 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 )}{315 \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}+\frac{\left (4 \left (140+111 a+21 a^2\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{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 )}{315 \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac{3}{8} (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4} \left (1+(-1+x)^2\right )+\frac{1}{4} \left (1+(-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}-\frac{4 \left (140+111 a+21 a^2\right ) \left (1-\sqrt{4+a}\right ) \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) (1-x)}{315 \sqrt{3+a-2 (1-x)^2-(1-x)^4}}-\frac{2}{315} \left (2 (80+27 a)+3 (20+7 a) (1-x)^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{63} \left (15+7 (-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)-\frac{4 (3+a) (100+33 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 )}{315 \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{1}{4} \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 (4 \left (140+111 a+21 a^2\right ) \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 )}{315 \sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac{3}{8} (4+a) \sqrt{3+a-2 (1-x)^2-(1-x)^4} \left (1+(-1+x)^2\right )+\frac{1}{4} \left (1+(-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}-\frac{4 \left (140+111 a+21 a^2\right ) \left (1-\sqrt{4+a}\right ) \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) (1-x)}{315 \sqrt{3+a-2 (1-x)^2-(1-x)^4}}-\frac{2}{315} \left (2 (80+27 a)+3 (20+7 a) (1-x)^2\right ) \sqrt{3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{63} \left (15+7 (-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac{3}{8} (4+a)^2 \tan ^{-1}\left (\frac{1+(-1+x)^2}{\sqrt{3+a-2 (1-x)^2-(1-x)^4}}\right )+\frac{4 \left (140+111 a+21 a^2\right ) \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 )}{315 \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) (100+33 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 )}{315 \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 = 6.16396, size = 8500, normalized size = 14.53 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]
[Out]
Result too large to show
________________________________________________________________________________________
Maple [B] time = 0.019, size = 2733, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}
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)^(3/2),x)
[Out]
-1/9*x^7*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+19/36*x^6*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-163/126*x^5*(-x^4+4*x^3-8*x^2
+a+8*x)^(1/2)+71/42*x^4*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(11/45*a-16/63)*x^3*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(-13
/120*a-5/18)*x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(9/140*a+23/63)*x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(107/252*a+10
1/63)*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-(-(9/140*a+23/63)*a-107/63*a-404/63)*((-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))-(-2*(-13/120*a-5/18)*a+827/315*a+76/9)
*((-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)))+(a^2-3*(11/45*a-16/63
)*a+68/105*a+16/9)*((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)
________________________________________________________________________________________
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^{2}\,{d x} \end{align*}
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)^(3/2),x, algorithm="maxima")
[Out]
integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(3/2)*x^2, x)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (x^{6} - 4 \, x^{5} + 8 \, x^{4} - a x^{2} - 8 \, x^{3}\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^2*(-x^4+4*x^3-8*x^2+a+8*x)^(3/2),x, algorithm="fricas")
[Out]
integral(-(x^6 - 4*x^5 + 8*x^4 - a*x^2 - 8*x^3)*sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x), x)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \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**2*(-x**4+4*x**3-8*x**2+a+8*x)**(3/2),x)
[Out]
Integral(x**2*(a - x**4 + 4*x**3 - 8*x**2 + 8*x)**(3/2), x)
________________________________________________________________________________________
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^{2}\,{d x} \end{align*}
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)^(3/2),x, algorithm="giac")
[Out]
integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(3/2)*x^2, x)