\(\int \frac {d+e x^4}{x (a+b x^4+c x^8)^2} \, dx\) [92]

Optimal result

Integrand size = 25, antiderivative size = 149 \[ \int \frac {d+e x^4}{x \left (a+b x^4+c x^8\right )^2} \, dx=\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x^4}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+\frac {\left (b^3 d-6 a b c d+4 a^2 c e\right ) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a^2 \left (b^2-4 a c\right )^{3/2}}+\frac {d \log (x)}{a^2}-\frac {d \log \left (a+b x^4+c x^8\right )}{8 a^2} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.16 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.24 \[ \int \frac {d+e x^4}{x \left (a+b x^4+c x^8\right )^2} \, dx=\frac {\frac {a \left (b^2 d+b \left (-a e+c d x^4\right )-2 a c \left (d+e x^4\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+4 d \log (x)-\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^3 d \log (x-\text {$\#$1})-5 a b c d \log (x-\text {$\#$1})+2 a^2 c e \log (x-\text {$\#$1})+b^2 c d \log (x-\text {$\#$1}) \text {$\#$1}^4-4 a c^2 d \log (x-\text {$\#$1}) \text {$\#$1}^4}{b+2 c \text {$\#$1}^4}\&\right ]}{b^2-4 a c}}{4 a^2} \] Input:

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

Output:

((a*(b^2*d + b*(-(a*e) + c*d*x^4) - 2*a*c*(d + e*x^4)))/((b^2 - 4*a*c)*(a 
+ b*x^4 + c*x^8)) + 4*d*Log[x] - RootSum[a + b*#1^4 + c*#1^8 & , (b^3*d*Lo 
g[x - #1] - 5*a*b*c*d*Log[x - #1] + 2*a^2*c*e*Log[x - #1] + b^2*c*d*Log[x 
- #1]*#1^4 - 4*a*c^2*d*Log[x - #1]*#1^4)/(b + 2*c*#1^4) & ]/(b^2 - 4*a*c)) 
/(4*a^2)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1802, 1235, 25, 1200, 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.

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

\(\Big \downarrow \) 1802

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1200

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \left (\frac {\frac {\text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right ) \left (4 a^2 c e-6 a b c d+b^3 d\right )}{a \sqrt {b^2-4 a c}}+\frac {d \log \left (x^4\right ) \left (b^2-4 a c\right )}{a}-\frac {d \left (b^2-4 a c\right ) \log \left (a+b x^4+c x^8\right )}{2 a}}{a \left (b^2-4 a c\right )}+\frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\right )\)

Input:

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

Output:

((b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^4)/(a*(b^2 - 4*a*c)*(a + b*x 
^4 + c*x^8)) + (((b^3*d - 6*a*b*c*d + 4*a^2*c*e)*ArcTanh[(b + 2*c*x^4)/Sqr 
t[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)*d*Log[x^4])/a - (( 
b^2 - 4*a*c)*d*Log[a + b*x^4 + c*x^8])/(2*a))/(a*(b^2 - 4*a*c)))/4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1200

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

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

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 = 0.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.42

method	result	size
default	\(\frac {\frac {\frac {a c \left (2 a e -b d \right ) x^{4}}{4 a c -b^{2}}+\frac {a \left (a b e +2 a c d -d \,b^{2}\right )}{4 a c -b^{2}}}{2 c \,x^{8}+2 b \,x^{4}+2 a}+\frac {\frac {\left (-4 a \,c^{2} d +b^{2} c d \right ) \ln \left (c \,x^{8}+b \,x^{4}+a \right )}{2 c}+\frac {2 \left (2 a^{2} c e -5 a b c d +b^{3} d -\frac {\left (-4 a \,c^{2} d +b^{2} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{8 a c -2 b^{2}}}{2 a^{2}}+\frac {d \ln \left (x \right )}{a^{2}}\)	\(211\)
risch	\(\frac {\frac {c \left (2 a e -b d \right ) x^{4}}{4 a \left (4 a c -b^{2}\right )}+\frac {a b e +2 a c d -d \,b^{2}}{4 \left (4 a c -b^{2}\right ) a}}{c \,x^{8}+b \,x^{4}+a}+\frac {d \ln \left (x \right )}{a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{5} c^{3}-48 a^{4} b^{2} c^{2}+12 a^{3} b^{4} c -a^{2} b^{6}\right ) \textit {\_Z}^{2}+\left (64 a^{3} c^{3} d -48 a^{2} b^{2} c^{2} d +12 a \,b^{4} c d -b^{6} d \right ) \textit {\_Z} +4 a^{2} c^{2} e^{2}-12 a b \,c^{2} d e +16 a \,c^{3} d^{2}+2 b^{3} c d e -3 b^{2} c^{2} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (288 a^{5} c^{3}-224 a^{4} b^{2} c^{2}+58 a^{3} b^{4} c -5 a^{2} b^{6}\right ) \textit {\_R}^{2}+\left (-8 a^{3} b \,c^{2} e +144 a^{3} c^{3} d +2 a^{2} b^{3} c e -68 a^{2} b^{2} c^{2} d +8 a \,b^{4} c d \right ) \textit {\_R} +16 a^{2} c^{2} e^{2}-16 a b \,c^{2} d e +4 b^{2} c^{2} d^{2}\right ) x^{4}+\left (-16 a^{5} b \,c^{2}+8 a^{4} b^{3} c -a^{3} b^{5}\right ) \textit {\_R}^{2}+\left (-8 a^{4} c^{2} e +2 a^{3} b^{2} c e +68 a^{3} b \,c^{2} d -33 a^{2} b^{3} c d +4 a \,b^{5} d \right ) \textit {\_R} +32 a^{2} c^{2} d e -8 a \,b^{2} c d e -16 a b \,c^{2} d^{2}+4 b^{3} c \,d^{2}\right )\right )}{4}\)	\(473\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (139) = 278\).

Time = 9.13 (sec) , antiderivative size = 989, normalized size of antiderivative = 6.64 \[ \int \frac {d+e x^4}{x \left (a+b x^4+c x^8\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/8*(2*((a*b^3*c - 4*a^2*b*c^2)*d - 2*(a^2*b^2*c - 4*a^3*c^2)*e)*x^4 - (( 
4*a^2*c^2*e + (b^3*c - 6*a*b*c^2)*d)*x^8 + 4*a^3*c*e + (4*a^2*b*c*e + (b^4 
 - 6*a*b^2*c)*d)*x^4 + (a*b^3 - 6*a^2*b*c)*d)*sqrt(b^2 - 4*a*c)*log((2*c^2 
*x^8 + 2*b*c*x^4 + b^2 - 2*a*c - (2*c*x^4 + b)*sqrt(b^2 - 4*a*c))/(c*x^8 + 
 b*x^4 + a)) + 2*(a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*d - 2*(a^2*b^3 - 4*a^3* 
b*c)*e - ((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^8 + (b^5 - 8*a*b^3*c + 16 
*a^2*b*c^2)*d*x^4 + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log(c*x^8 + b*x^ 
4 + a) + 8*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^8 + (b^5 - 8*a*b^3*c + 
16*a^2*b*c^2)*d*x^4 + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log(x))/((a^2* 
b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*x^8 + a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c 
^2 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^4), 1/8*(2*((a*b^3*c - 4*a^2 
*b*c^2)*d - 2*(a^2*b^2*c - 4*a^3*c^2)*e)*x^4 + 2*((4*a^2*c^2*e + (b^3*c - 
6*a*b*c^2)*d)*x^8 + 4*a^3*c*e + (4*a^2*b*c*e + (b^4 - 6*a*b^2*c)*d)*x^4 + 
(a*b^3 - 6*a^2*b*c)*d)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^4 + b)*sqrt(-b^2 
+ 4*a*c)/(b^2 - 4*a*c)) + 2*(a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*d - 2*(a^2*b 
^3 - 4*a^3*b*c)*e - ((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^8 + (b^5 - 8*a 
*b^3*c + 16*a^2*b*c^2)*d*x^4 + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log(c 
*x^8 + b*x^4 + a) + 8*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^8 + (b^5 - 8 
*a*b^3*c + 16*a^2*b*c^2)*d*x^4 + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log 
(x))/((a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*x^8 + a^3*b^4 - 8*a^4*b^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x^4+d)/x/(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 [A] (verification not implemented)

Time = 4.75 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.35 \[ \int \frac {d+e x^4}{x \left (a+b x^4+c x^8\right )^2} \, dx=-\frac {{\left (b^{3} d - 6 \, a b c d + 4 \, a^{2} c e\right )} \arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {d \log \left (c x^{8} + b x^{4} + a\right )}{8 \, a^{2}} + \frac {d \log \left (x^{4}\right )}{4 \, a^{2}} + \frac {b^{2} c d x^{8} - 4 \, a c^{2} d x^{8} + b^{3} d x^{4} - 2 \, a b c d x^{4} - 4 \, a^{2} c e x^{4} + 3 \, a b^{2} d - 8 \, a^{2} c d - 2 \, a^{2} b e}{8 \, {\left (c x^{8} + b x^{4} + a\right )} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} \] Input:

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

Output:

-1/4*(b^3*d - 6*a*b*c*d + 4*a^2*c*e)*arctan((2*c*x^4 + b)/sqrt(-b^2 + 4*a* 
c))/((a^2*b^2 - 4*a^3*c)*sqrt(-b^2 + 4*a*c)) - 1/8*d*log(c*x^8 + b*x^4 + a 
)/a^2 + 1/4*d*log(x^4)/a^2 + 1/8*(b^2*c*d*x^8 - 4*a*c^2*d*x^8 + b^3*d*x^4 
- 2*a*b*c*d*x^4 - 4*a^2*c*e*x^4 + 3*a*b^2*d - 8*a^2*c*d - 2*a^2*b*e)/((c*x 
^8 + b*x^4 + a)*(a^2*b^2 - 4*a^3*c))
 

Mupad [B] (verification not implemented)

Time = 85.15 (sec) , antiderivative size = 35718, normalized size of antiderivative = 239.72 \[ \int \frac {d+e x^4}{x \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((a*b*e - b^2*d + 2*a*c*d)/(4*a*(4*a*c - b^2)) + (c*x^4*(2*a*e - b*d))/(4* 
a*(4*a*c - b^2)))/(a + b*x^4 + c*x^8) - (log((((d + a^2*(-(b^3*d + 4*a^2*c 
*e - 6*a*b*c*d)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*(((d + a^2*(-(b^3*d + 4*a^ 
2*c*e - 6*a*b*c*d)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*(((d + a^2*(-(b^3*d + 4 
*a^2*c*e - 6*a*b*c*d)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*(((d + a^2*(-(b^3*d 
+ 4*a^2*c*e - 6*a*b*c*d)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*((32*b^3*c^4*(d + 
 a^2*(-(b^3*d + 4*a^2*c*e - 6*a*b*c*d)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*(a* 
b + 5*b^2*x^4 - 18*a*c*x^4))/a^2 - (256*b^3*c^4*(b^3*d + 2*a^2*c*e - 5*a*b 
*c*d))/(a*(4*a*c - b^2)) + (64*b^2*c^5*x^4*(7*b^3*d - 32*a*b^2*e + 108*a^2 
*c*e - 18*a*b*c*d))/(a*(4*a*c - b^2))))/(8*a^2) + (32*b^2*c^5*(2*a*e - b*d 
)*(8*b^3*d + 6*a^2*c*e - 35*a*b*c*d))/(a^2*(4*a*c - b^2)^2) + (16*b*c^6*x^ 
4*(2*a*e - b*d)*(13*b^3*d + 28*a*b^2*e - 108*a^2*c*e - 54*a*b*c*d))/(a^2*( 
4*a*c - b^2)^2)))/(8*a^2) - (16*b*c^6*(2*a*e - b*d)^2*(6*b^3*d + 2*a^2*c*e 
 - 25*a*b*c*d))/(a^3*(4*a*c - b^2)^3) + (4*c^7*x^4*(2*a*e - b*d)^2*(8*a*b^ 
2*e - 31*b^3*d + 36*a^2*c*e + 90*a*b*c*d))/(a^3*(4*a*c - b^2)^3)))/(8*a^2) 
 + (c^7*(2*a*e - b*d)^3*(16*b^3*d + 2*a^2*c*e - 65*a*b*c*d))/(a^4*(4*a*c - 
 b^2)^4) - (2*c^8*x^4*(2*a*e - b*d)^3*(11*a*b*e - 10*b^2*d + 18*a*c*d))/(a 
^4*(4*a*c - b^2)^4)))/(8*a^2) + (c^8*d*(2*a*e - b*d)^4)/(a^5*(4*a*c - b^2) 
^4) + (c^9*x^4*(2*a*e - b*d)^5)/(a^5*(4*a*c - b^2)^5))*(((d - a^2*(-(b^3*d 
 + 4*a^2*c*e - 6*a*b*c*d)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*(((d - a^2*(-...
 

Reduce [F]

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

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

Output:

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