3.8.81 \(\int (a+8 x-8 x^2+4 x^3-x^4)^{3/2} \, dx\) [781]

3.8.81.1 Optimal result

Integrand size = 24, antiderivative size = 452 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=-\frac {16 (7+2 a) \left (1-\sqrt {4+a}\right ) \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) (-1+x)}{35 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {2}{35} \left (13+5 a-3 (-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{7} \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac {16 (7+2 a) \left (1-\sqrt {4+a}\right ) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) E\left (\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{35 \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {4 (3+a) (16+5 a) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right ),-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{35 \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}} \]

output

1/7*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2)*(-1+x)-16/35*(7+2*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)+2/35*(1 
3+5*a-3*(-1+x)^2)*(-1+x)*(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)+4/35*(3+a)*(16+5* 
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)+16/35*(7+2*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)
 

3.8.81.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(6287\) vs. \(2(452)=904\).

Time = 16.10 (sec) , antiderivative size = 6287, normalized size of antiderivative = 13.91 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

3.8.81.3 Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2458, 1404, 27, 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.

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

\(\Big \downarrow \) 2458

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

\(\Big \downarrow \) 1404

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \int \left (-(x-1)^2+a+3\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {6}{7} \left (\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}-\frac {1}{15} \int -\frac {2 \left ((a+3) (5 a+16)-4 (2 a+7) (x-1)^2\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \left (\frac {2}{15} \int \frac {(a+3) (5 a+16)-4 (2 a+7) (x-1)^2}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \int \frac {(a+3) (5 a+16)-4 (2 a+7) (x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left ((a+3) (5 a+16) \int \frac {1}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)-4 (2 a+7) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) (5 a+16) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-4 (2 a+7) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) (5 a+16) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-4 (2 a+7) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\left (1-\sqrt {a+4}\right ) \int \frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\left (\frac {(x-1)^2}{\sqrt {a+4}+1}+1\right )^{3/2}}d(x-1)\right )\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) (5 a+16) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-4 (2 a+7) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\frac {\left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} E\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

input

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

output

((3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x))/7 + (6*(((13 + 5*a - 
3*(-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/15 + (2*Sq 
rt[1 + (-1 + x)^2/(1 - Sqrt[4 + a])]*Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a]) 
]*(-4*(7 + 2*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])*Sqr 
t[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])])/(Sq 
rt[(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)*(16 + 5*a)*Sqrt[1 + Sq 
rt[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])))/7
 

3.8.81.3.1 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 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 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 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 2458

Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp 
on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x 
- S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp 
on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P 
n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
 

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

Leaf count of result is larger than twice the leaf count of optimal. \(2654\) vs. \(2(506)=1012\).

Time = 5.24 (sec) , antiderivative size = 2655, normalized size of antiderivative = 5.87

method	result	size
default	\(\text {Expression too large to display}\)	\(2655\)
elliptic	\(\text {Expression too large to display}\)	\(2655\)
risch	\(\text {Expression too large to display}\)	\(3593\)

input

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

output

-1/7*x^5*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+5/7*x^4*(-x^4+4*x^3-8*x^2+a+8*x)^( 
1/2)-66/35*x^3*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+14/5*x^2*(-x^4+4*x^3-8*x^2+a 
+8*x)^(1/2)+(3/7*a-32/35)*x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(-3/7*a-4/7)*(- 
x^4+4*x^3-8*x^2+a+8*x)^(1/2)-(a^2-(3/7*a-32/35)*a+12/7*a+16/7)*((-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))-(64/35*a+32/5)*((-1-(a+ 
4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*((-(-1-(a+4)^(1/2))^(1/2)+(-1+(...
 

3.8.81.5 Fricas [F]

\[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} \,d x } \]

input

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

output

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

3.8.81.6 Sympy [F]

\[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int \left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {3}{2}}\, dx \]

input

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

output

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

3.8.81.7 Maxima [F]

\[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} \,d x } \]

input

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

output

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

3.8.81.8 Giac [F]

\[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} \,d x } \]

input

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

output

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

3.8.81.9 Mupad [F(-1)]

Timed out. \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int {\left (-x^4+4\,x^3-8\,x^2+8\,x+a\right )}^{3/2} \,d x \]

input

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

output

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