3.8.92 \(\int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx\) [792]

3.8.92.1 Optimal result

Integrand size = 28, antiderivative size = 485 \[ \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\frac {1}{2} \left (1+(-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}+\frac {2 (8+3 a) \left (1-\sqrt {4+a}\right ) \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) (-1+x)}{15 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {1}{15} \left (7+3 (-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{2} (4+a) \arctan \left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\right )-\frac {2 (8+3 a) \left (1-\sqrt {4+a}\right ) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) E\left (\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{15 \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {8 (3+a) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right ),-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{15 \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}} \]

output

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)
 

3.8.92.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(5647\) vs. \(2(485)=970\).

Time = 13.84 (sec) , antiderivative size = 5647, normalized size of antiderivative = 11.64 \[ \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

3.8.92.3 Rubi [A] (warning: unable to verify)

Time = 0.80 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.14, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {2459, 2006, 2202, 27, 1432, 1087, 1092, 217, 1490, 27, 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.

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

\(\Big \downarrow \) 2459

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

\(\Big \downarrow \) 2006

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

\(\Big \downarrow \) 2202

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1432

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1490

\(\displaystyle -\frac {1}{15} \int -\frac {2 \left ((3 a+8) (x-1)^2+4 (a+3)\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{15} \int \frac {(3 a+8) (x-1)^2+4 (a+3)}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \int \frac {(3 a+8) (x-1)^2+4 (a+3)}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (4 (a+3) \int \frac {1}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)+(3 a+8) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left ((3 a+8) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)+\frac {4 (a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left ((3 a+8) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\left (1-\sqrt {a+4}\right ) \int \frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\left (\frac {(x-1)^2}{\sqrt {a+4}+1}+1\right )^{3/2}}d(x-1)\right )+\frac {4 (a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {4 (a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}+(3 a+8) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\frac {\left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} E\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\)

input

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

output

((7 + 3*(-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/15 + 
 (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*Sqrt[1 + (-1 + x)^2/(1 - Sqrt[4 
+ a])]*Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])]*((8 + 3*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 - Sqrt[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])])) + (4*(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]))]*Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])])))/(15*Sqrt 
[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])
 

3.8.92.3.1 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 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 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> 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] + Simp[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 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 2006

Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], 
b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px 
, x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P 
x, x], 1] && NeQ[Coeff[Px, x, 0], 0] &&  !MatchQ[Px, (a_.)*(v_)^Expon[Px, x 
] /; FreeQ[a, x] && LinearQ[v, x]]
 

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])
 

3.8.92.4 Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2581\) vs. \(2(535)=1070\).

Time = 4.11 (sec) , antiderivative size = 2582, normalized size of antiderivative = 5.32

method	result	size
default	\(\text {Expression too large to display}\)	\(2582\)
elliptic	\(\text {Expression too large to display}\)	\(2582\)
risch	\(\text {Expression too large to display}\)	\(3044\)

input

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

output

1/5*x^3*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-1/10*x^2*(-x^4+4*x^3-8*x^2+a+8*x)^( 
1/2)+1/15*x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+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)*E 
llipticF(((-(-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))^(...
 

3.8.92.5 Fricas [F]

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

input

integrate(x^2*(-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^2, x)
 

3.8.92.6 Sympy [F]

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

input

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

output

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

3.8.92.7 Maxima [F]

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

input

integrate(x^2*(-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^2, x)
 

3.8.92.8 Giac [F]

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

input

integrate(x^2*(-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^2, x)
 

3.8.92.9 Mupad [F(-1)]

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

input

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

output

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