\(\int x \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx\) [34]

Optimal result

Integrand size = 26, antiderivative size = 407 \[ \int x \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\frac {1}{4} \sqrt {3+a-2 (1-x)^2-(1-x)^4} \left (1+(-1+x)^2\right )-\frac {1}{3} \sqrt {3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac {1}{4} (4+a) \arctan \left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (1-x)^2-(1-x)^4}}\right )+\frac {2 \sqrt {-1+\sqrt {4+a}} \left (1+\sqrt {4+a}\right ) \sqrt {1+\frac {(1-x)^2}{1-\sqrt {4+a}}} \sqrt {1+\frac {(1-x)^2}{1+\sqrt {4+a}}} E\left (\arcsin \left (\frac {1-x}{\sqrt {-1+\sqrt {4+a}}}\right )|\frac {1-\sqrt {4+a}}{1+\sqrt {4+a}}\right )}{3 \sqrt {3+a-2 (1-x)^2-(1-x)^4}}-\frac {2 \sqrt {-1+\sqrt {4+a}} \left (4+a+\sqrt {4+a}\right ) \sqrt {1+\frac {(1-x)^2}{1-\sqrt {4+a}}} \sqrt {1+\frac {(1-x)^2}{1+\sqrt {4+a}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-x}{\sqrt {-1+\sqrt {4+a}}}\right ),\frac {1-\sqrt {4+a}}{1+\sqrt {4+a}}\right )}{3 \sqrt {3+a-2 (1-x)^2-(1-x)^4}} \] Output:

1/4*(3+a-2*(1-x)^2-(1-x)^4)^(1/2)*(1+(-1+x)^2)-1/3*(3+a-2*(1-x)^2-(1-x)^4) 
^(1/2)*(1-x)+1/4*(4+a)*arctan((1+(-1+x)^2)/(3+a-2*(1-x)^2-(1-x)^4)^(1/2))+ 
2/3*(-1+(4+a)^(1/2))^(1/2)*(1+(4+a)^(1/2))*(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-(4+a)^(1/2))/(1+(4+a)^(1/2)))^(1/2))/(3+a-2*(1-x)^2-(1-x)^4)^(1/2)-2 
/3*(-1+(4+a)^(1/2))^(1/2)*(4+a+(4+a)^(1/2))*(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-(4+a)^(1/2))/(1+(4+a)^(1/2)))^(1/2))/(3+a-2*(1-x)^2-(1-x)^4)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(4389\) vs. \(2(407)=814\).

Time = 16.41 (sec) , antiderivative size = 4389, normalized size of antiderivative = 10.78 \[ \int x \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\text {Result too large to show} \] Input:

Integrate[x*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]
 

Output:

(1/6 - x/6 + x^2/4)*Sqrt[a - x*(-8 + 8*x - 4*x^2 + x^3)] + (Sqrt[a - x*(-8 
 + 8*x - 4*x^2 + x^3)]*((-8*(-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 - Sqr 
t[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sq 
rt[(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 - Sq 
rt[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 + 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 - ...
 

Rubi [A] (warning: unable to verify)

Time = 1.23 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.33, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2459, 2202, 1404, 27, 1432, 1087, 1092, 217, 1514, 406, 320, 388, 313}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2459

\(\displaystyle \int x \sqrt {a-(x-1)^4-2 (x-1)^2+3}d(x-1)\)

\(\Big \downarrow \) 2202

\(\displaystyle \int \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\int \sqrt {-(x-1)^4-2 (x-1)^2+a+3} (x-1)d(x-1)\)

\(\Big \downarrow \) 1404

\(\displaystyle \frac {1}{3} \int \frac {2 \left (-(x-1)^2+a+3\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\int \sqrt {-(x-1)^4-2 (x-1)^2+a+3} (x-1)d(x-1)+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \int \frac {-(x-1)^2+a+3}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\int \sqrt {-(x-1)^4-2 (x-1)^2+a+3} (x-1)d(x-1)+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 1432

\(\displaystyle \frac {2}{3} \int \frac {-(x-1)^2+a+3}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{2} \int \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)^2+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} (a+4) \int \frac {1}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)^2+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {2}{3} \int \frac {-(x-1)^2+a+3}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{2} \left ((a+4) \int \frac {1}{-(x-1)^4-4}d\left (-\frac {2 x}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {2}{3} \int \frac {-(x-1)^2+a+3}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {2}{3} \int \frac {-(x-1)^2+a+3}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{2} \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \int \frac {-(x-1)^2+a+3}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)}{3 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left ((a+3) \int \frac {1}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)-\int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{3 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{3 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\left (1-\sqrt {a+4}\right ) \int \frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\left (\frac {(x-1)^2}{\sqrt {a+4}+1}+1\right )^{3/2}}d(x-1)+\frac {(a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )}{3 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}+\frac {\left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} E\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )}{3 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{3} (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

Input:

Int[x*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]
 

Output:

(Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/3 + ((Sqrt[3 + a - 2*(- 
1 + x)^2 - (-1 + x)^4]*x)/2 + ((4 + a)*ArcTan[x/Sqrt[3 + a - 2*(-1 + x)^2 
- (-1 + x)^4]])/2)/2 + (2*Sqrt[1 + (-1 + x)^2/(1 - Sqrt[4 + a])]*Sqrt[1 + 
(-1 + x)^2/(1 + Sqrt[4 + a])]*(-(((1 - Sqrt[4 + a])*Sqrt[1 + (-1 + x)^2/(1 
 - Sqrt[4 + a])]*(-1 + x))/Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])]) + ((1 - 
 Sqrt[4 + a])*Sqrt[1 + Sqrt[4 + a]]*Sqrt[1 + (-1 + x)^2/(1 - Sqrt[4 + a])] 
*EllipticE[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - S 
qrt[4 + a])])/(Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 
+ Sqrt[4 + a]))]*Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])]) + ((3 + a)*Sqrt[1 
 + Sqrt[4 + a]]*Sqrt[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])])/(Sqrt[ 
(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*Sqr 
t[1 + (-1 + x)^2/(1 + Sqrt[4 + a])])))/(3*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 
+ x)^4])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 313

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

rule 320

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

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[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 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

rule 1087

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] - Simp[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] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

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

rule 1404

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] + Simp[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 1432

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

rule 1514

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[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)/(Sqrt[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 2202

Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n 
 = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b 
*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 
1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] 
 &&  !PolyQ[Pn, x^2]
 

rule 2459

Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 
]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x 
 -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial 
Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - 
> x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ 
[Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] &&  !(MonomialQ[Qx, x] 
&& IGtQ[p, 0])
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2550\) vs. \(2(345)=690\).

Time = 7.24 (sec) , antiderivative size = 2551, normalized size of antiderivative = 6.27

method	result	size
default	\(\text {Expression too large to display}\)	\(2551\)
elliptic	\(\text {Expression too large to display}\)	\(2551\)
risch	\(\text {Expression too large to display}\)	\(3034\)

Input:

int(x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/4*x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-1/6*x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2 
)+1/6*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-(1/6*a-2/3)*((-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))-(1/2*a+10/3)*((-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*...
 

Fricas [F]

\[ \int x \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\int { \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x \,d x } \] Input:

integrate(x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)*x, x)
 

Sympy [F]

\[ \int x \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\int x \sqrt {a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x}\, dx \] Input:

integrate(x*(-x**4+4*x**3-8*x**2+a+8*x)**(1/2),x)
 

Output:

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

Maxima [F]

\[ \int x \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\int { \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x \,d x } \] Input:

integrate(x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int x \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\int { \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x \,d x } \] Input:

integrate(x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\int x\,\sqrt {-x^4+4\,x^3-8\,x^2+8\,x+a} \,d x \] Input:

int(x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2),x)
 

Output:

int(x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x)
 

Reduce [F]

\[ \int x \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{2}}{4}-\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x}{6}+\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{18}+\frac {\left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right ) a}{6}-\frac {2 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right )}{9}-\frac {2 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{3}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right )}{9}+\frac {\left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right ) a}{2}+\frac {22 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right )}{9} \] Input:

int(x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x)
 

Output:

(9*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x**2 - 6*sqrt(a - x**4 + 4*x**3 
- 8*x**2 + 8*x)*x + 2*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x) + 6*int(sqrt( 
a - x**4 + 4*x**3 - 8*x**2 + 8*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x)*a 
- 8*int(sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)/(a - x**4 + 4*x**3 - 8*x**2 
 + 8*x),x) - 8*int((sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x**3)/(a - x**4 
 + 4*x**3 - 8*x**2 + 8*x),x) + 18*int((sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8 
*x)*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x)*a + 88*int((sqrt(a - x**4 + 4 
*x**3 - 8*x**2 + 8*x)*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x))/36