\(\int \frac {1}{x (d+e x^4) (a+b x^4+c x^8)^2} \, dx\) [108]

Optimal result

Integrand size = 27, antiderivative size = 391 \[ \int \frac {1}{x \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^4}{4 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^4+c x^8\right )}-\frac {c \left (b c d-b^2 e+2 a c e\right ) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{2 a \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )}+\frac {\left (b^3 d e^2+2 a^2 c e^3+b c d \left (c d^2+2 a e^2\right )-2 b^2 \left (c d^2 e+a e^3\right )\right ) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a^2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}+\frac {\log (x)}{a^2 d}-\frac {e^4 \log \left (d+e x^4\right )}{4 d \left (c d^2-b d e+a e^2\right )^2}-\frac {(c d-b e) \left (c d^2-e (b d-2 a e)\right ) \log \left (a+b x^4+c x^8\right )}{8 a^2 \left (c d^2-b d e+a e^2\right )^2} \] Output:

1/4*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e+c*(2*a*c*e-b^2*e+b*c*d)*x^4)/a/(-4* 
a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^8+b*x^4+a)-1/2*c*(2*a*c*e-b^2*e+b*c*d)*a 
rctanh((2*c*x^4+b)/(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c 
*d^2)+1/4*(b^3*d*e^2+2*a^2*c*e^3+b*c*d*(2*a*e^2+c*d^2)-2*b^2*(a*e^3+c*d^2* 
e))*arctanh((2*c*x^4+b)/(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(1/2)/(a*e^2- 
b*d*e+c*d^2)^2+ln(x)/a^2/d-1/4*e^4*ln(e*x^4+d)/d/(a*e^2-b*d*e+c*d^2)^2-1/8 
*(-b*e+c*d)*(c*d^2-e*(-2*a*e+b*d))*ln(c*x^8+b*x^4+a)/a^2/(a*e^2-b*d*e+c*d^ 
2)^2
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.58 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {1}{4} \left (\frac {b^3 e-b c \left (3 a e+c d x^4\right )+2 a c^2 \left (d-e x^4\right )+b^2 c \left (-d+e x^4\right )}{a \left (-b^2+4 a c\right ) \left (c d^2+e (-b d+a e)\right ) \left (a+b x^4+c x^8\right )}+\frac {4 \log (x)}{a^2 d}-\frac {e^4 \log \left (d+e x^4\right )}{d \left (c d^2+e (-b d+a e)\right )^2}+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^3 c^2 d^3 \log (x-\text {$\#$1})-5 a b c^3 d^3 \log (x-\text {$\#$1})-2 b^4 c d^2 e \log (x-\text {$\#$1})+10 a b^2 c^2 d^2 e \log (x-\text {$\#$1})-2 a^2 c^3 d^2 e \log (x-\text {$\#$1})+b^5 d e^2 \log (x-\text {$\#$1})-3 a b^3 c d e^2 \log (x-\text {$\#$1})-7 a^2 b c^2 d e^2 \log (x-\text {$\#$1})-2 a b^4 e^3 \log (x-\text {$\#$1})+10 a^2 b^2 c e^3 \log (x-\text {$\#$1})-6 a^3 c^2 e^3 \log (x-\text {$\#$1})+b^2 c^3 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^4-4 a c^4 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^4-2 b^3 c^2 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+8 a b c^3 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+b^4 c d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-2 a b^2 c^2 d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-8 a^2 c^3 d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-2 a b^3 c e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4+8 a^2 b c^2 e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{b+2 c \text {$\#$1}^4}\&\right ]}{a^2 \left (-b^2+4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2}\right ) \] Input:

Integrate[1/(x*(d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

((b^3*e - b*c*(3*a*e + c*d*x^4) + 2*a*c^2*(d - e*x^4) + b^2*c*(-d + e*x^4) 
)/(a*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*(a + b*x^4 + c*x^8)) + (4*L 
og[x])/(a^2*d) - (e^4*Log[d + e*x^4])/(d*(c*d^2 + e*(-(b*d) + a*e))^2) + R 
ootSum[a + b*#1^4 + c*#1^8 & , (b^3*c^2*d^3*Log[x - #1] - 5*a*b*c^3*d^3*Lo 
g[x - #1] - 2*b^4*c*d^2*e*Log[x - #1] + 10*a*b^2*c^2*d^2*e*Log[x - #1] - 2 
*a^2*c^3*d^2*e*Log[x - #1] + b^5*d*e^2*Log[x - #1] - 3*a*b^3*c*d*e^2*Log[x 
 - #1] - 7*a^2*b*c^2*d*e^2*Log[x - #1] - 2*a*b^4*e^3*Log[x - #1] + 10*a^2* 
b^2*c*e^3*Log[x - #1] - 6*a^3*c^2*e^3*Log[x - #1] + b^2*c^3*d^3*Log[x - #1 
]*#1^4 - 4*a*c^4*d^3*Log[x - #1]*#1^4 - 2*b^3*c^2*d^2*e*Log[x - #1]*#1^4 + 
 8*a*b*c^3*d^2*e*Log[x - #1]*#1^4 + b^4*c*d*e^2*Log[x - #1]*#1^4 - 2*a*b^2 
*c^2*d*e^2*Log[x - #1]*#1^4 - 8*a^2*c^3*d*e^2*Log[x - #1]*#1^4 - 2*a*b^3*c 
*e^3*Log[x - #1]*#1^4 + 8*a^2*b*c^2*e^3*Log[x - #1]*#1^4)/(b + 2*c*#1^4) & 
 ]/(a^2*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2))/4
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1802, 1289, 2009}

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.

\(\displaystyle \int \frac {1}{x \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx\)

\(\Big \downarrow \) 1802

\(\displaystyle \frac {1}{4} \int \frac {1}{x^4 \left (e x^4+d\right ) \left (c x^8+b x^4+a\right )^2}dx^4\)

\(\Big \downarrow \) 1289

\(\displaystyle \frac {1}{4} \int \left (-\frac {e^5}{d \left (c d^2-b e d+a e^2\right )^2 \left (e x^4+d\right )}+\frac {-c (c d-b e) \left (c d^2-e (b d-2 a e)\right ) x^4-a^2 c e^3-b^3 d e^2-b c d \left (c d^2+2 a e^2\right )+2 b^2 \left (a e^3+c d^2 e\right )}{a^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^8+b x^4+a\right )}+\frac {-c (c d-b e) x^4-b c d+b^2 e-a c e}{a \left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )^2}+\frac {1}{a^2 d x^4}\right )dx^4\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \left (\frac {\text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right ) \left (2 a^2 c e^3-2 b^2 \left (a e^3+c d^2 e\right )+b c d \left (2 a e^2+c d^2\right )+b^3 d e^2\right )}{a^2 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}-\frac {(c d-b e) \left (c d^2-e (b d-2 a e)\right ) \log \left (a+b x^4+c x^8\right )}{2 a^2 \left (a e^2-b d e+c d^2\right )^2}+\frac {\log \left (x^4\right )}{a^2 d}-\frac {2 c \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right ) \left (2 a c e+b^2 (-e)+b c d\right )}{a \left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {c x^4 \left (2 a c e+b^2 (-e)+b c d\right )+3 a b c e-2 a c^2 d-b^3 e+b^2 c d}{a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right ) \left (a e^2-b d e+c d^2\right )}-\frac {e^4 \log \left (d+e x^4\right )}{d \left (a e^2-b d e+c d^2\right )^2}\right )\)

Input:

Int[1/(x*(d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

((b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x^ 
4)/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x^4 + c*x^8)) - (2*c*(b 
*c*d - b^2*e + 2*a*c*e)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(a*(b^2 
- 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)) + ((b^3*d*e^2 + 2*a^2*c*e^3 + b*c* 
d*(c*d^2 + 2*a*e^2) - 2*b^2*(c*d^2*e + a*e^3))*ArcTanh[(b + 2*c*x^4)/Sqrt[ 
b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^2) + Log[x^4 
]/(a^2*d) - (e^4*Log[d + e*x^4])/(d*(c*d^2 - b*d*e + a*e^2)^2) - ((c*d - b 
*e)*(c*d^2 - e*(b*d - 2*a*e))*Log[a + b*x^4 + c*x^8])/(2*a^2*(c*d^2 - b*d* 
e + a*e^2)^2))/4
 

Defintions of rubi rules used

rule 1289

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( 
IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
 

rule 1802

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( 
e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1 
)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, 
c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 57.37 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.72

method	result	size
default	\(-\frac {\frac {\frac {a c \left (2 a^{2} c \,e^{3}-a \,b^{2} e^{3}-a b c d \,e^{2}+2 a \,c^{2} d^{2} e +b^{3} d \,e^{2}-2 b^{2} c \,d^{2} e +b \,c^{2} d^{3}\right ) x^{4}}{4 a c -b^{2}}+\frac {a \left (3 a^{2} b c \,e^{3}-2 a^{2} c^{2} e^{2} d -a \,b^{3} e^{3}-2 a \,b^{2} c d \,e^{2}+5 a b \,c^{2} d^{2} e -2 a \,c^{3} d^{3}+b^{4} d \,e^{2}-2 b^{3} c \,d^{2} e +b^{2} c^{2} d^{3}\right )}{4 a c -b^{2}}}{2 c \,x^{8}+2 b \,x^{4}+2 a}+\frac {\frac {\left (-8 a^{2} b \,c^{2} e^{3}+8 a^{2} c^{3} d \,e^{2}+2 a \,b^{3} c \,e^{3}+2 a \,b^{2} c^{2} d \,e^{2}-8 a b \,c^{3} d^{2} e +4 a \,c^{4} d^{3}-b^{4} c d \,e^{2}+2 b^{3} c^{2} d^{2} e -b^{2} c^{3} d^{3}\right ) \ln \left (c \,x^{8}+b \,x^{4}+a \right )}{2 c}+\frac {2 \left (6 a^{3} c^{2} e^{3}-10 a^{2} b^{2} c \,e^{3}+7 a^{2} b \,c^{2} d \,e^{2}+2 a^{2} c^{3} d^{2} e +2 a \,b^{4} e^{3}+3 a \,b^{3} c d \,e^{2}-10 a \,b^{2} c^{2} d^{2} e +5 a b \,c^{3} d^{3}-b^{5} d \,e^{2}+2 b^{4} c \,d^{2} e -b^{3} c^{2} d^{3}-\frac {\left (-8 a^{2} b \,c^{2} e^{3}+8 a^{2} c^{3} d \,e^{2}+2 a \,b^{3} c \,e^{3}+2 a \,b^{2} c^{2} d \,e^{2}-8 a b \,c^{3} d^{2} e +4 a \,c^{4} d^{3}-b^{4} c d \,e^{2}+2 b^{3} c^{2} d^{2} e -b^{2} c^{3} d^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{8 a c -2 b^{2}}}{2 a^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {e^{4} \ln \left (x^{4} e +d \right )}{4 d \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {\ln \left (x \right )}{a^{2} d}\)	\(672\)
risch	\(\text {Expression too large to display}\)	\(3679\)

Input:

int(1/x/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/2/a^2/(a*e^2-b*d*e+c*d^2)^2*(1/2*(a*c*(2*a^2*c*e^3-a*b^2*e^3-a*b*c*d*e^ 
2+2*a*c^2*d^2*e+b^3*d*e^2-2*b^2*c*d^2*e+b*c^2*d^3)/(4*a*c-b^2)*x^4+a*(3*a^ 
2*b*c*e^3-2*a^2*c^2*d*e^2-a*b^3*e^3-2*a*b^2*c*d*e^2+5*a*b*c^2*d^2*e-2*a*c^ 
3*d^3+b^4*d*e^2-2*b^3*c*d^2*e+b^2*c^2*d^3)/(4*a*c-b^2))/(c*x^8+b*x^4+a)+1/ 
2/(4*a*c-b^2)*(1/2*(-8*a^2*b*c^2*e^3+8*a^2*c^3*d*e^2+2*a*b^3*c*e^3+2*a*b^2 
*c^2*d*e^2-8*a*b*c^3*d^2*e+4*a*c^4*d^3-b^4*c*d*e^2+2*b^3*c^2*d^2*e-b^2*c^3 
*d^3)/c*ln(c*x^8+b*x^4+a)+2*(6*a^3*c^2*e^3-10*a^2*b^2*c*e^3+7*a^2*b*c^2*d* 
e^2+2*a^2*c^3*d^2*e+2*a*b^4*e^3+3*a*b^3*c*d*e^2-10*a*b^2*c^2*d^2*e+5*a*b*c 
^3*d^3-b^5*d*e^2+2*b^4*c*d^2*e-b^3*c^2*d^3-1/2*(-8*a^2*b*c^2*e^3+8*a^2*c^3 
*d*e^2+2*a*b^3*c*e^3+2*a*b^2*c^2*d*e^2-8*a*b*c^3*d^2*e+4*a*c^4*d^3-b^4*c*d 
*e^2+2*b^3*c^2*d^2*e-b^2*c^3*d^3)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b 
)/(4*a*c-b^2)^(1/2))))-1/4*e^4*ln(e*x^4+d)/d/(a*e^2-b*d*e+c*d^2)^2+ln(x)/a 
^2/d
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/x/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/x/(e*x**4+d)/(c*x**8+b*x**4+a)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (373) = 746\).

Time = 4.46 (sec) , antiderivative size = 1019, normalized size of antiderivative = 2.61 \[ \int \frac {1}{x \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/x/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
 

Output:

-1/4*e^5*log(abs(e*x^4 + d))/(c^2*d^5*e - 2*b*c*d^4*e^2 + b^2*d^3*e^3 + 2* 
a*c*d^3*e^3 - 2*a*b*d^2*e^4 + a^2*d*e^5) - 1/8*(c^2*d^3 - 2*b*c*d^2*e + b^ 
2*d*e^2 + 2*a*c*d*e^2 - 2*a*b*e^3)*log(c*x^8 + b*x^4 + a)/(a^2*c^2*d^4 - 2 
*a^2*b*c*d^3*e + a^2*b^2*d^2*e^2 + 2*a^3*c*d^2*e^2 - 2*a^3*b*d*e^3 + a^4*e 
^4) - 1/4*(b^3*c^2*d^3 - 6*a*b*c^3*d^3 - 2*b^4*c*d^2*e + 12*a*b^2*c^2*d^2* 
e - 4*a^2*c^3*d^2*e + b^5*d*e^2 - 4*a*b^3*c*d*e^2 - 6*a^2*b*c^2*d*e^2 - 2* 
a*b^4*e^3 + 12*a^2*b^2*c*e^3 - 12*a^3*c^2*e^3)*arctan((2*c*x^4 + b)/sqrt(- 
b^2 + 4*a*c))/((a^2*b^2*c^2*d^4 - 4*a^3*c^3*d^4 - 2*a^2*b^3*c*d^3*e + 8*a^ 
3*b*c^2*d^3*e + a^2*b^4*d^2*e^2 - 2*a^3*b^2*c*d^2*e^2 - 8*a^4*c^2*d^2*e^2 
- 2*a^3*b^3*d*e^3 + 8*a^4*b*c*d*e^3 + a^4*b^2*e^4 - 4*a^5*c*e^4)*sqrt(-b^2 
 + 4*a*c)) + 1/8*(b^2*c^3*d^3*x^8 - 4*a*c^4*d^3*x^8 - 2*b^3*c^2*d^2*e*x^8 
+ 8*a*b*c^3*d^2*e*x^8 + b^4*c*d*e^2*x^8 - 2*a*b^2*c^2*d*e^2*x^8 - 8*a^2*c^ 
3*d*e^2*x^8 - 2*a*b^3*c*e^3*x^8 + 8*a^2*b*c^2*e^3*x^8 + b^3*c^2*d^3*x^4 - 
2*a*b*c^3*d^3*x^4 - 2*b^4*c*d^2*e*x^4 + 4*a*b^2*c^2*d^2*e*x^4 + 4*a^2*c^3* 
d^2*e*x^4 + b^5*d*e^2*x^4 - 10*a^2*b*c^2*d*e^2*x^4 - 2*a*b^4*e^3*x^4 + 6*a 
^2*b^2*c*e^3*x^4 + 4*a^3*c^2*e^3*x^4 + 3*a*b^2*c^2*d^3 - 8*a^2*c^3*d^3 - 6 
*a*b^3*c*d^2*e + 18*a^2*b*c^2*d^2*e + 3*a*b^4*d*e^2 - 6*a^2*b^2*c*d*e^2 - 
12*a^3*c^2*d*e^2 - 4*a^2*b^3*e^3 + 14*a^3*b*c*e^3)/((a^2*b^2*c^2*d^4 - 4*a 
^3*c^3*d^4 - 2*a^2*b^3*c*d^3*e + 8*a^3*b*c^2*d^3*e + a^2*b^4*d^2*e^2 - 2*a 
^3*b^2*c*d^2*e^2 - 8*a^4*c^2*d^2*e^2 - 2*a^3*b^3*d*e^3 + 8*a^4*b*c*d*e^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Hanged} \] Input:

int(1/(x*(d + e*x^4)*(a + b*x^4 + c*x^8)^2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1}{x \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {1}{x \left (e \,x^{4}+d \right ) \left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:

int(1/x/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
 

Output:

int(1/x/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)