\(\int \frac {d+e x^4}{(a+b x^4+c x^8)^2} \, dx\) [103]

Optimal result

Integrand size = 22, antiderivative size = 548 \[ \int \frac {d+e x^4}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}-\frac {c^{3/4} \left (3 b d-6 a e-\frac {3 b^2 d-28 a c d+8 a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} a \left (b^2-4 a c\right ) \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {c^{3/4} \left (3 b d-6 a e+\frac {3 b^2 d-28 a c d+8 a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} a \left (b^2-4 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {c^{3/4} \left (3 b d-6 a e-\frac {3 b^2 d-28 a c d+8 a b e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} a \left (b^2-4 a c\right ) \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {c^{3/4} \left (3 b d-6 a e+\frac {3 b^2 d-28 a c d+8 a b e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} a \left (b^2-4 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}} \] Output:

1/4*x*(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)-1/16*c^(3/4)*(3*b*d-6*a*e-(8*a*b*e-28*a*c*d+3*b^2*d)/(-4*a*c+b^2)^(1/2 
))*arctan(2^(1/4)*c^(1/4)*x/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/a/(-4*a 
*c+b^2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-1/16*c^(3/4)*(3*b*d-6*a*e+(8*a*b*e-2 
8*a*c*d+3*b^2*d)/(-4*a*c+b^2)^(1/2))*arctan(2^(1/4)*c^(1/4)*x/(-b+(-4*a*c+ 
b^2)^(1/2))^(1/4))*2^(3/4)/a/(-4*a*c+b^2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)-1/ 
16*c^(3/4)*(3*b*d-6*a*e-(8*a*b*e-28*a*c*d+3*b^2*d)/(-4*a*c+b^2)^(1/2))*arc 
tanh(2^(1/4)*c^(1/4)*x/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/a/(-4*a*c+b^ 
2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-1/16*c^(3/4)*(3*b*d-6*a*e+(8*a*b*e-28*a*c 
*d+3*b^2*d)/(-4*a*c+b^2)^(1/2))*arctanh(2^(1/4)*c^(1/4)*x/(-b+(-4*a*c+b^2) 
^(1/2))^(1/4))*2^(3/4)/a/(-4*a*c+b^2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)
 

Mathematica [C] (verified)

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

Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.31 \[ \int \frac {d+e x^4}{\left (a+b x^4+c x^8\right )^2} \, dx=-\frac {\frac {4 x \left (b^2 d+b \left (-a e+c d x^4\right )-2 a c \left (d+e x^4\right )\right )}{a+b x^4+c x^8}+\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {3 b^2 d \log (x-\text {$\#$1})-14 a c d \log (x-\text {$\#$1})+a b e \log (x-\text {$\#$1})+3 b c d \log (x-\text {$\#$1}) \text {$\#$1}^4-6 a c e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{16 a \left (-b^2+4 a c\right )} \] Input:

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

Output:

-1/16*((4*x*(b^2*d + b*(-(a*e) + c*d*x^4) - 2*a*c*(d + e*x^4)))/(a + b*x^4 
 + c*x^8) + RootSum[a + b*#1^4 + c*#1^8 & , (3*b^2*d*Log[x - #1] - 14*a*c* 
d*Log[x - #1] + a*b*e*Log[x - #1] + 3*b*c*d*Log[x - #1]*#1^4 - 6*a*c*e*Log 
[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(a*(-b^2 + 4*a*c))
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1760, 25, 1752, 756, 218, 221}

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.

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

\(\Big \downarrow \) 1760

\(\displaystyle \frac {x \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}-\frac {\int -\frac {3 c (b d-2 a e) x^4+3 b^2 d-14 a c d+a b e}{c x^8+b x^4+a}dx}{4 a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {3 c (b d-2 a e) x^4+3 b^2 d-14 a c d+a b e}{c x^8+b x^4+a}dx}{4 a \left (b^2-4 a c\right )}+\frac {x \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 1752

\(\displaystyle \frac {\frac {1}{2} c \left (\frac {8 a b e-28 a c d+3 b^2 d}{\sqrt {b^2-4 a c}}+3 (b d-2 a e)\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (3 (b d-2 a e)-\frac {8 a b e-28 a c d+3 b^2 d}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{4 a \left (b^2-4 a c\right )}+\frac {x \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {\frac {1}{2} c \left (3 (b d-2 a e)-\frac {8 a b e-28 a c d+3 b^2 d}{\sqrt {b^2-4 a c}}\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {-b-\sqrt {b^2-4 a c}}}dx}{\sqrt {-\sqrt {b^2-4 a c}-b}}\right )+\frac {1}{2} c \left (\frac {8 a b e-28 a c d+3 b^2 d}{\sqrt {b^2-4 a c}}+3 (b d-2 a e)\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {\sqrt {b^2-4 a c}-b}}dx}{\sqrt {\sqrt {b^2-4 a c}-b}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {x \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {1}{2} c \left (3 (b d-2 a e)-\frac {8 a b e-28 a c d+3 b^2 d}{\sqrt {b^2-4 a c}}\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )+\frac {1}{2} c \left (\frac {8 a b e-28 a c d+3 b^2 d}{\sqrt {b^2-4 a c}}+3 (b d-2 a e)\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {x \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{2} c \left (3 (b d-2 a e)-\frac {8 a b e-28 a c d+3 b^2 d}{\sqrt {b^2-4 a c}}\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )+\frac {1}{2} c \left (\frac {8 a b e-28 a c d+3 b^2 d}{\sqrt {b^2-4 a c}}+3 (b d-2 a e)\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {x \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

Input:

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

Output:

(x*(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^4))/(4*a*(b^2 - 4*a*c)*(a 
+ b*x^4 + c*x^8)) + ((c*(3*(b*d - 2*a*e) - (3*b^2*d - 28*a*c*d + 8*a*b*e)/ 
Sqrt[b^2 - 4*a*c])*(-(ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^ 
(1/4)]/(2^(1/4)*c^(1/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTanh[(2^(1/4 
)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b - Sqrt[b 
^2 - 4*a*c])^(3/4))))/2 + (c*(3*(b*d - 2*a*e) + (3*b^2*d - 28*a*c*d + 8*a* 
b*e)/Sqrt[b^2 - 4*a*c])*(-(ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a 
*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTanh[(2 
^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b + S 
qrt[b^2 - 4*a*c])^(3/4))))/2)/(4*a*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

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

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 1752

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x 
_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) 
   Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   I 
nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 
, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 
 - 4*a*c] ||  !IGtQ[n/2, 0])
 

rule 1760

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p 
_), x_Symbol] :> Simp[(-x)*(d*b^2 - a*b*e - 2*a*c*d + (b*d - 2*a*e)*c*x^n)* 
((a + b*x^n + c*x^(2*n))^(p + 1)/(a*n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/ 
(a*n*(p + 1)*(b^2 - 4*a*c))   Int[Simp[(n*p + n + 1)*d*b^2 - a*b*e - 2*a*c* 
d*(2*n*p + 2*n + 1) + (2*n*p + 3*n + 1)*(d*b - 2*a*e)*c*x^n, x]*(a + b*x^n 
+ c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n 
] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]
 

Maple [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.31

method	result	size
default	\(\frac {\frac {c \left (2 a e -b d \right ) x^{5}}{4 a \left (4 a c -b^{2}\right )}+\frac {\left (a b e +2 a c d -d \,b^{2}\right ) x}{4 \left (4 a c -b^{2}\right ) a}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (3 c \left (2 a e -b d \right ) \textit {\_R}^{4}-a b e +14 a c d -3 d \,b^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{16 a \left (4 a c -b^{2}\right )}\)	\(168\)
risch	\(\frac {\frac {c \left (2 a e -b d \right ) x^{5}}{4 a \left (4 a c -b^{2}\right )}+\frac {\left (a b e +2 a c d -d \,b^{2}\right ) x}{4 \left (4 a c -b^{2}\right ) a}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\frac {3 c \left (2 a e -b d \right ) \textit {\_R}^{4}}{4 a c -b^{2}}-\frac {a b e -14 a c d +3 d \,b^{2}}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{16 a}\)	\(182\)

Input:

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

Output:

(1/4*c*(2*a*e-b*d)/a/(4*a*c-b^2)*x^5+1/4*(a*b*e+2*a*c*d-b^2*d)/(4*a*c-b^2) 
/a*x)/(c*x^8+b*x^4+a)+1/16/a/(4*a*c-b^2)*sum((3*c*(2*a*e-b*d)*_R^4-a*b*e+1 
4*a*c*d-3*d*b^2)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/4*((b*c*d - 2*a*c*e)*x^5 - (a*b*e - (b^2 - 2*a*c)*d)*x)/((a*b^2*c - 4*a^ 
2*c^2)*x^8 + (a*b^3 - 4*a^2*b*c)*x^4 + a^2*b^2 - 4*a^3*c) - 1/4*integrate( 
-(3*(b*c*d - 2*a*c*e)*x^4 + a*b*e + (3*b^2 - 14*a*c)*d)/(c*x^8 + b*x^4 + a 
), x)/(a*b^2 - 4*a^2*c)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 31.21 (sec) , antiderivative size = 128217, normalized size of antiderivative = 233.97 \[ \int \frac {d+e x^4}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((19548*a*b^6*c^8*d^5 - 891*b^8*c^7*d^5 - 537824*a^4*c^11*d^5 - 116 
64*a^6*b*c^8*e^5 + 54432*a^6*c^9*d*e^4 + 567*b^9*c^6*d^4*e - 155358*a^2*b^ 
4*c^9*d^5 + 510384*a^3*b^2*c^10*d^5 + 7*a^4*b^5*c^6*e^5 - 2232*a^5*b^3*c^7 
*e^5 + 197568*a^5*c^10*d^3*e^2 - 8802*a^2*b^6*c^7*d^3*e^2 + 378*a^2*b^7*c^ 
6*d^2*e^3 - 3060*a^3*b^4*c^8*d^3*e^2 + 174*a^3*b^5*c^7*d^2*e^3 + 157696*a^ 
4*b^2*c^9*d^3*e^2 + 22192*a^4*b^3*c^8*d^2*e^3 - 13959*a*b^7*c^7*d^4*e + 22 
5008*a^4*b*c^10*d^4*e + 756*a*b^8*c^6*d^3*e^2 + 118071*a^2*b^5*c^8*d^4*e - 
 367080*a^3*b^3*c^9*d^4*e + 84*a^3*b^6*c^6*d*e^4 - 6166*a^4*b^4*c^7*d*e^4 
- 320544*a^5*b*c^9*d^2*e^3 + 60912*a^5*b^2*c^8*d*e^4)/(64*(a^4*b^8 + 256*a 
^8*c^4 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^3)) + ((x*(32212254 
72*a^12*b*c^12*e^2 - 202937204736*a^11*b*c^13*d^2 + 589824*a^2*b^19*c^4*d^ 
2 - 22609920*a^3*b^17*c^5*d^2 + 382271488*a^4*b^15*c^6*d^2 - 3741057024*a^ 
5*b^13*c^7*d^2 + 23350738944*a^6*b^11*c^8*d^2 - 96380911616*a^7*b^9*c^9*d^ 
2 + 262982860800*a^8*b^7*c^10*d^2 - 457212690432*a^9*b^5*c^11*d^2 + 459293 
065216*a^10*b^3*c^12*d^2 + 65536*a^4*b^17*c^4*e^2 - 983040*a^5*b^15*c^5*e^ 
2 + 2359296*a^6*b^13*c^6*e^2 + 38797312*a^7*b^11*c^7*e^2 - 314572800*a^8*b 
^9*c^8*e^2 + 855638016*a^9*b^7*c^9*e^2 - 335544320*a^10*b^5*c^10*e^2 - 241 
5919104*a^11*b^3*c^11*e^2 + 90194313216*a^12*c^13*d*e + 393216*a^3*b^18*c^ 
4*d*e - 10485760*a^4*b^16*c^5*d*e + 107741184*a^5*b^14*c^6*d*e - 449839104 
*a^6*b^12*c^7*d*e - 507510784*a^7*b^10*c^8*d*e + 13941866496*a^8*b^8*c^...
 

Reduce [F]

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

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

Output:

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