\(\int \frac {1}{(a+8 x-8 x^2+4 x^3-x^4)^{5/2}} \, dx\) [72]

Optimal result

Integrand size = 24, antiderivative size = 460 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=-\frac {\left (104+47 a+5 a^2+4 (7+2 a) (1-x)^2\right ) (1-x)}{12 (3+a)^2 (4+a)^2 \sqrt {3+a-2 (1-x)^2-(1-x)^4}}-\frac {\left (5+a+(-1+x)^2\right ) (1-x)}{6 \left (12+7 a+a^2\right ) \left (3+a-2 (1-x)^2-(1-x)^4\right )^{3/2}}+\frac {(7+2 a) \sqrt {-1+\sqrt {4+a}} \left (1+\sqrt {4+a}\right ) \sqrt {1+\frac {(1-x)^2}{1-\sqrt {4+a}}} \sqrt {1+\frac {(1-x)^2}{1+\sqrt {4+a}}} E\left (\arcsin \left (\frac {1-x}{\sqrt {-1+\sqrt {4+a}}}\right )|\frac {1-\sqrt {4+a}}{1+\sqrt {4+a}}\right )}{3 (3+a)^2 (4+a)^2 \sqrt {3+a-2 (1-x)^2-(1-x)^4}}-\frac {\sqrt {-1+\sqrt {4+a}} \left (76+5 a^2+28 \sqrt {4+a}+a \left (39+8 \sqrt {4+a}\right )\right ) \sqrt {1+\frac {(1-x)^2}{1-\sqrt {4+a}}} \sqrt {1+\frac {(1-x)^2}{1+\sqrt {4+a}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-x}{\sqrt {-1+\sqrt {4+a}}}\right ),\frac {1-\sqrt {4+a}}{1+\sqrt {4+a}}\right )}{12 (3+a)^2 (4+a)^2 \sqrt {3+a-2 (1-x)^2-(1-x)^4}} \] Output:

-1/12*(104+47*a+5*a^2+4*(7+2*a)*(1-x)^2)*(1-x)/(3+a)^2/(4+a)^2/(3+a-2*(1-x 
)^2-(1-x)^4)^(1/2)-1/6*(5+a+(-1+x)^2)*(1-x)/(a^2+7*a+12)/(3+a-2*(1-x)^2-(1 
-x)^4)^(3/2)+1/3*(7+2*a)*(-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/(4 
+a)^2/(3+a-2*(1-x)^2-(1-x)^4)^(1/2)-1/12*(-1+(4+a)^(1/2))^(1/2)*(76+5*a^2+ 
28*(4+a)^(1/2)+a*(39+8*(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/(4+a)^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. \(6386\) vs. \(2(460)=920\).

Time = 16.22 (sec) , antiderivative size = 6386, normalized size of antiderivative = 13.88 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-5/2),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.30, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2458, 1405, 27, 1492, 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.

\(\displaystyle \int \frac {1}{\left (a-x^4+4 x^3-8 x^2+8 x\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2458

\(\displaystyle \int \frac {1}{\left (a-(x-1)^4-2 (x-1)^2+3\right )^{5/2}}d(x-1)\)

\(\Big \downarrow \) 1405

\(\displaystyle \frac {(x-1) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}-\frac {\int -\frac {2 \left (3 (x-1)^2+5 a+19\right )}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}}d(x-1)}{12 \left (a^2+7 a+12\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 (x-1)^2+5 a+19}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}}d(x-1)}{6 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {\frac {(x-1) \left (5 a^2+4 (2 a+7) (x-1)^2+47 a+104\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}-\frac {\int -\frac {2 \left ((a+3) (5 a+16)-4 (2 a+7) (x-1)^2\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)}{4 \left (a^2+7 a+12\right )}}{6 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(a+3) (5 a+16)-4 (2 a+7) (x-1)^2}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)}{2 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (5 a^2+4 (2 a+7) (x-1)^2+47 a+104\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}}{6 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {\frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \int \frac {(a+3) (5 a+16)-4 (2 a+7) (x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1) \left (5 a^2+4 (2 a+7) (x-1)^2+47 a+104\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}}{6 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {\frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left ((a+3) (5 a+16) \int \frac {1}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)-4 (2 a+7) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1) \left (5 a^2+4 (2 a+7) (x-1)^2+47 a+104\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}}{6 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) (5 a+16) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-4 (2 a+7) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1) \left (5 a^2+4 (2 a+7) (x-1)^2+47 a+104\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}}{6 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) (5 a+16) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-4 (2 a+7) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\left (1-\sqrt {a+4}\right ) \int \frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\left (\frac {(x-1)^2}{\sqrt {a+4}+1}+1\right )^{3/2}}d(x-1)\right )\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1) \left (5 a^2+4 (2 a+7) (x-1)^2+47 a+104\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}}{6 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) (5 a+16) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-4 (2 a+7) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\frac {\left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} E\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1) \left (5 a^2+4 (2 a+7) (x-1)^2+47 a+104\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}}{6 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\)

Input:

Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-5/2),x]
 

Output:

((5 + a + (-1 + x)^2)*(-1 + x))/(6*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 
- (-1 + x)^4)^(3/2)) + (((104 + 47*a + 5*a^2 + 4*(7 + 2*a)*(-1 + x)^2)*(-1 
 + x))/(2*(12 + 7*a + a^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + (Sqr 
t[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])*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])])/(Sqr 
t[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*S 
qrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])])) + ((3 + a)*(16 + 5*a)*Sqrt[1 + Sqr 
t[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])])))/(2*(12 + 7*a + a^2)*Sqrt[3 + a - 2*(-1 + 
x)^2 - (-1 + x)^4]))/(6*(12 + 7*a + a^2))
 

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 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) 
), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
 

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*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] && 
 LtQ[p, -1] && 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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2756\) vs. \(2(400)=800\).

Time = 0.92 (sec) , antiderivative size = 2757, normalized size of antiderivative = 5.99

method	result	size
default	\(\text {Expression too large to display}\)	\(2757\)
elliptic	\(\text {Expression too large to display}\)	\(2757\)

Input:

int(1/(-x^4+4*x^3-8*x^2+a+8*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

(1/6/(a^2+7*a+12)*x^3-1/2/(a^2+7*a+12)*x^2+1/6*(a+8)/(a^2+7*a+12)*x-1/6*(6 
+a)/(a^2+7*a+12))*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)/(x^4-4*x^3+8*x^2-a-8*x)^2 
+2*(1/6*(7+2*a)/(a^2+7*a+12)^2*x^3-1/2*(7+2*a)/(a^2+7*a+12)^2*x^2+1/24*(5* 
a^2+71*a+188)/(a^2+7*a+12)^2*x-1/24*(5*a^2+55*a+132)/(a^2+7*a+12)^2)/(-x^4 
+4*x^3-8*x^2+a+8*x)^(1/2)-(1/6*(5*a^2+47*a+104)/(a^2+7*a+12)^2-1/12*(5*a^2 
+71*a+188)/(a^2+7*a+12)^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-(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...
 

Fricas [F]

\[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(-x^4+4*x^3-8*x^2+a+8*x)^(5/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)/(x^12 - 12*x^11 + 72*x^10 - 
 3*(a - 256)*x^8 - 280*x^9 + 24*(a - 64)*x^7 - 32*(3*a - 70)*x^6 + 48*(5*a 
 - 48)*x^5 + 3*(a^2 - 128*a + 512)*x^4 - 4*(3*a^2 - 96*a + 128)*x^3 - a^3 
- 24*a^2*x + 24*(a^2 - 8*a)*x^2), x)
 

Sympy [F]

\[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(-x**4+4*x**3-8*x**2+a+8*x)**(5/2),x)
 

Output:

Integral((a - x**4 + 4*x**3 - 8*x**2 + 8*x)**(-5/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(-x^4+4*x^3-8*x^2+a+8*x)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(-5/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(-x^4+4*x^3-8*x^2+a+8*x)^(5/2),x, algorithm="giac")
 

Output:

integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(-5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (-x^4+4\,x^3-8\,x^2+8\,x+a\right )}^{5/2}} \,d x \] Input:

int(1/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(5/2),x)
 

Output:

int(1/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{-x^{12}+12 x^{11}-72 x^{10}+3 a \,x^{8}+280 x^{9}-24 a \,x^{7}-768 x^{8}+96 a \,x^{6}+1536 x^{7}-3 a^{2} x^{4}-240 a \,x^{5}-2240 x^{6}+12 a^{2} x^{3}+384 a \,x^{4}+2304 x^{5}-24 a^{2} x^{2}-384 a \,x^{3}-1536 x^{4}+a^{3}+24 a^{2} x +192 a \,x^{2}+512 x^{3}}d x \] Input:

int(1/(-x^4+4*x^3-8*x^2+a+8*x)^(5/2),x)
 

Output:

int(sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)/(a**3 - 3*a**2*x**4 + 12*a**2*x 
**3 - 24*a**2*x**2 + 24*a**2*x + 3*a*x**8 - 24*a*x**7 + 96*a*x**6 - 240*a* 
x**5 + 384*a*x**4 - 384*a*x**3 + 192*a*x**2 - x**12 + 12*x**11 - 72*x**10 
+ 280*x**9 - 768*x**8 + 1536*x**7 - 2240*x**6 + 2304*x**5 - 1536*x**4 + 51 
2*x**3),x)