3.792 \(\int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx\)
Optimal. Leaf size=485 \[ \frac {1}{2} \left ((x-1)^2+1\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {2 (3 a+8) \left (1-\sqrt {a+4}\right ) (x-1) \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} (a+4) \tan ^{-1}\left (\frac {(x-1)^2+1}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {8 (a+3) \sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{15 \sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}}-\frac {2 (3 a+8) \left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{15 \sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}} \]
[Out]
1/2*(4+a)*arctan((1+(-1+x)^2)/(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2))+2/15*(8+3*a)*(-1+x)*(1+(-1+x)^2/(1-(4+a)^(1/2))
)*(1-(4+a)^(1/2))/(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)+1/2*(1+(-1+x)^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)+1/15*(7+3*(
-1+x)^2)*(-1+x)*(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)+8/15*(3+a)*(1/(1+(-1+x)^2/(1+(4+a)^(1/2))))^(1/2)*(1+(-1+x)^2/
(1+(4+a)^(1/2)))^(1/2)*EllipticF((-1+x)/(1+(4+a)^(1/2))^(1/2)/(1+(-1+x)^2/(1+(4+a)^(1/2)))^(1/2),(-2*(4+a)^(1/
2)/(1-(4+a)^(1/2)))^(1/2))*(1+(-1+x)^2/(1-(4+a)^(1/2)))*(1+(4+a)^(1/2))^(1/2)/(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)/
((1+(-1+x)^2/(1-(4+a)^(1/2)))/(1+(-1+x)^2/(1+(4+a)^(1/2))))^(1/2)-2/15*(8+3*a)*(1/(1+(-1+x)^2/(1+(4+a)^(1/2)))
)^(1/2)*(1+(-1+x)^2/(1+(4+a)^(1/2)))^(1/2)*EllipticE((-1+x)/(1+(4+a)^(1/2))^(1/2)/(1+(-1+x)^2/(1+(4+a)^(1/2)))
^(1/2),(-2*(4+a)^(1/2)/(1-(4+a)^(1/2)))^(1/2))*(1+(-1+x)^2/(1-(4+a)^(1/2)))*(1-(4+a)^(1/2))*(1+(4+a)^(1/2))^(1
/2)/(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)/((1+(-1+x)^2/(1-(4+a)^(1/2)))/(1+(-1+x)^2/(1+(4+a)^(1/2))))^(1/2)
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Rubi [A] time = 0.53, antiderivative size = 485, normalized size of antiderivative = 1.00,
number of steps used = 13, 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}
\[ \frac {1}{2} \left ((x-1)^2+1\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {2 (3 a+8) \left (1-\sqrt {a+4}\right ) (x-1) \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} (a+4) \tan ^{-1}\left (\frac {(x-1)^2+1}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {8 (a+3) \sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{15 \sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}}-\frac {2 (3 a+8) \left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{15 \sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}} \]
Antiderivative was successfully verified.
[In]
Int[x^2*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]
[Out]
((1 + (-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])/2 + (2*(8 + 3*a)*(1 - Sqrt[4 + a])*(1 + (-1 + x)^2/
(1 - Sqrt[4 + a]))*(-1 + x))/(15*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + ((7 + 3*(-1 + x)^2)*Sqrt[3 + a - 2
*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/15 + ((4 + a)*ArcTan[(1 + (-1 + x)^2)/Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)
^4]])/2 - (2*(8 + 3*a)*(1 - Sqrt[4 + a])*Sqrt[1 + Sqrt[4 + a]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*EllipticE[Ar
cTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(15*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]) + (8*(3 + a)*Sqrt[1 + Sq
rt[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])])/(15*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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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 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 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 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 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 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 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 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]
Rubi steps
\begin {align*} \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx &=\operatorname {Subst}\left (\int (1+x)^2 \sqrt {3+a-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=\operatorname {Subst}\left (\int 2 x \sqrt {3+a-2 x^2-x^4} \, dx,x,-1+x\right )+\operatorname {Subst}\left (\int \left (1+x^2\right ) \sqrt {3+a-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=\frac {1}{15} \left (7+3 (-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4} (-1+x)-\frac {1}{15} \operatorname {Subst}\left (\int \frac {-8 (3+a)-2 (8+3 a) x^2}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )+2 \operatorname {Subst}\left (\int x \sqrt {3+a-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=\frac {1}{15} \left (7+3 (-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4} (-1+x)-\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 {-8 (3+a)-2 (8+3 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 )}{15 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\operatorname {Subst}\left (\int \sqrt {3+a-2 x-x^2} \, dx,x,(-1+x)^2\right )\\ &=\frac {1}{2} \left (1+(-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}+\frac {1}{15} \left (7+3 (-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{2} (4+a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+a-2 x-x^2}} \, dx,x,(-1+x)^2\right )+\frac {\left (8 (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 )}{15 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {\left (2 (8+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 {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 )}{15 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac {1}{2} \left (1+(-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}-\frac {2 (8+3 a) \left (1-\sqrt {4+a}\right ) \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) (1-x)}{15 \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {1}{15} \left (7+3 (-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4} (-1+x)-\frac {8 (3+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 )}{15 \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}}+(4+a) \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 (2 (8+3 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 )}{15 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac {1}{2} \left (1+(-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}-\frac {2 (8+3 a) \left (1-\sqrt {4+a}\right ) \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) (1-x)}{15 \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {1}{15} \left (7+3 (-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{2} (4+a) \tan ^{-1}\left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (1-x)^2-(1-x)^4}}\right )+\frac {2 (8+3 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 )}{15 \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 {8 (3+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 )}{15 \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*}
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Mathematica [B] time = 6.12, size = 5647, normalized size = 11.64 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
[In]
Integrate[x^2*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]
[Out]
Result too large to show
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x^{2}, x\right ) \]
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)^(1/2),x, algorithm="fricas")
[Out]
integral(sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)*x^2, x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x^{2}\,{d x} \]
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)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)*x^2, x)
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maple [B] time = 0.03, size = 2582, normalized size = 5.32 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-x^4+4*x^3-8*x^2+a+8*x)^(1/2)*x^2,x)
[Out]
1/5*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)*x^3-1/10*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)*x^2+1/15*(-x^4+4*x^3-8*x^2+a+8*x)^(
1/2)*x+1/3*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-(-1/15*a-4/3)*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*((-(-1
-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1
/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2)*(x-1+(-1+(a+4)^(1/2))^(1/2))^2*(-2*(-1+(a+4)^(1/2))^(1/2)*(x-1
-(-1-(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2)*(
-2*(-1+(a+4)^(1/2))^(1/2)*(x-1+(-1-(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(
-1+(a+4)^(1/2))^(1/2)))^(1/2)/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-1+(a+4)^(1/2))^(1/2)/(-(x-1-(
-1+(a+4)^(1/2))^(1/2))*(x-1+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1-(a+4)^(1/2))^(1/2))*(x-1+(-1-(a+4)^(1/2))^(1/2)))
^(1/2)*EllipticF(((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1
/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2),((-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/
2))^(1/2))*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/((
-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)))^(1/2))-(1/5*a+28/15)*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(
1/2))*((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-
(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2)*(x-1+(-1+(a+4)^(1/2))^(1/2))^2*(-2*(-1+(a+4)^(1/2)
)^(1/2)*(x-1-(-1-(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/
2)))^(1/2)*(-2*(-1+(a+4)^(1/2))^(1/2)*(x-1+(-1-(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(
1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2)/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-1+(a+4)^(1/2))^(1
/2)/(-(x-1-(-1+(a+4)^(1/2))^(1/2))*(x-1+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1-(a+4)^(1/2))^(1/2))*(x-1+(-1-(a+4)^(1
/2))^(1/2)))^(1/2)*((1-(-1+(a+4)^(1/2))^(1/2))*EllipticF(((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-
1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2)
,((-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)
^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)))^(1/2))+2*(-1+(a+4)^(1/2
))^(1/2)*EllipticPi(((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)
^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2),(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(
1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2)),((-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)
)*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/((-1-(a+4)^
(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)))^(1/2)))+(2/5*a+16/15)*((x-1-(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1-(a+4)^(1/2)
)^(1/2))*(x-1+(-1-(a+4)^(1/2))^(1/2))+((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*((-(-1-(a+4)^(1/2))^(1/2
)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(
-1+(a+4)^(1/2))^(1/2)))^(1/2)*(x-1+(-1+(a+4)^(1/2))^(1/2))^2*(-2*(-1+(a+4)^(1/2))^(1/2)*(x-1-(-1-(a+4)^(1/2))^
(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2)*(-2*(-1+(a+4)^(1/2)
)^(1/2)*(x-1+(-1-(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1
/2)))^(1/2)*(-1/2*((1-(-1+(a+4)^(1/2))^(1/2))*(1+(-1+(a+4)^(1/2))^(1/2))-(1-(-1-(a+4)^(1/2))^(1/2))*(1+(-1+(a+
4)^(1/2))^(1/2))+(1-(-1-(a+4)^(1/2))^(1/2))*(1-(-1+(a+4)^(1/2))^(1/2))+(1-(-1+(a+4)^(1/2))^(1/2))^2)/(-(-1-(a+
4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-1+(a+4)^(1/2))^(1/2)*EllipticF(((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(
1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2
))^(1/2)))^(1/2),((-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1
/2))/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)))^(1/2))-
1/2*(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*EllipticE(((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2
))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))
^(1/2),((-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-(-1
-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)))^(1/2))/(-1+(a+4)^
(1/2))^(1/2)-4/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*EllipticPi(((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)
^(1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1
/2))^(1/2)))^(1/2),((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1
/2)),((-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-(-1-(
a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)))^(1/2))))/(-(x-1-(-1
+(a+4)^(1/2))^(1/2))*(x-1+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1-(a+4)^(1/2))^(1/2))*(x-1+(-1-(a+4)^(1/2))^(1/2)))^(
1/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x^{2}\,{d x} \]
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)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)*x^2, x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\sqrt {-x^4+4\,x^3-8\,x^2+8\,x+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2),x)
[Out]
int(x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x}\, dx \]
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)**(1/2),x)
[Out]
Integral(x**2*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x), x)
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