\(\int \frac {d+e x^4}{x^2 (a+b x^4+c x^8)^2} \, dx\) [104]

Optimal result

Integrand size = 25, antiderivative size = 659 \[ \int \frac {d+e x^4}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=-\frac {5 b^2 d-18 a c d-a b e}{4 a^2 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x^4}{4 a \left (b^2-4 a c\right ) x \left (a+b x^4+c x^8\right )}-\frac {\sqrt [4]{c} \left (5 b^2 d-18 a c d-a b e-\frac {5 b^3 d-28 a b c d-a b^2 e+20 a^2 c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (5 b^2 d-18 a c d-a b e+\frac {5 b^3 d-28 a b c d-a b^2 e+20 a^2 c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^2 d-18 a c d-a b e-\frac {5 b^3 d-28 a b c d-a b^2 e+20 a^2 c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^2 d-18 a c d-a b e+\frac {5 b^3 d-28 a b c d-a b^2 e+20 a^2 c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \] Output:

-1/4*(-a*b*e-18*a*c*d+5*b^2*d)/a^2/(-4*a*c+b^2)/x+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)/x/(c*x^8+b*x^4+a)-1/16*c^(1/4)*(5*b^2* 
d-18*a*c*d-a*b*e-(20*a^2*c*e-a*b^2*e-28*a*b*c*d+5*b^3*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^(1/4)/a^2/(-4 
*a*c+b^2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)-1/16*c^(1/4)*(5*b^2*d-18*a*c*d-a*b 
*e+(20*a^2*c*e-a*b^2*e-28*a*b*c*d+5*b^3*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^(1/4)/a^2/(-4*a*c+b^2)/(-b+ 
(-4*a*c+b^2)^(1/2))^(1/4)+1/16*c^(1/4)*(5*b^2*d-18*a*c*d-a*b*e-(20*a^2*c*e 
-a*b^2*e-28*a*b*c*d+5*b^3*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^(1/4)/a^2/(-4*a*c+b^2)/(-b-(-4*a*c+b^2)^ 
(1/2))^(1/4)+1/16*c^(1/4)*(5*b^2*d-18*a*c*d-a*b*e+(20*a^2*c*e-a*b^2*e-28*a 
*b*c*d+5*b^3*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^(1/4)/a^2/(-4*a*c+b^2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)
 

Mathematica [C] (verified)

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

Time = 0.34 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.37 \[ \int \frac {d+e x^4}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=-\frac {\frac {16 d}{x}+\frac {4 x^3 \left (b^3 d+2 a c \left (a e-c d x^4\right )+b^2 \left (-a e+c d x^4\right )-a b c \left (3 d+e x^4\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {5 b^3 d \log (x-\text {$\#$1})-23 a b c d \log (x-\text {$\#$1})-a b^2 e \log (x-\text {$\#$1})+10 a^2 c e \log (x-\text {$\#$1})+5 b^2 c d \log (x-\text {$\#$1}) \text {$\#$1}^4-18 a c^2 d \log (x-\text {$\#$1}) \text {$\#$1}^4-a b c e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{b^2-4 a c}}{16 a^2} \] Input:

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

Output:

-1/16*((16*d)/x + (4*x^3*(b^3*d + 2*a*c*(a*e - c*d*x^4) + b^2*(-(a*e) + c* 
d*x^4) - a*b*c*(3*d + e*x^4)))/((b^2 - 4*a*c)*(a + b*x^4 + c*x^8)) + RootS 
um[a + b*#1^4 + c*#1^8 & , (5*b^3*d*Log[x - #1] - 23*a*b*c*d*Log[x - #1] - 
 a*b^2*e*Log[x - #1] + 10*a^2*c*e*Log[x - #1] + 5*b^2*c*d*Log[x - #1]*#1^4 
 - 18*a*c^2*d*Log[x - #1]*#1^4 - a*b*c*e*Log[x - #1]*#1^4)/(b*#1 + 2*c*#1^ 
5) & ]/(b^2 - 4*a*c))/a^2
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 510, normalized size of antiderivative = 0.77, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1824, 25, 1828, 1834, 27, 827, 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.

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

\(\Big \downarrow \) 1824

\(\displaystyle \frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}-\frac {\int -\frac {5 c (b d-2 a e) x^4+5 b^2 d-18 a c d-a b e}{x^2 \left (c x^8+b x^4+a\right )}dx}{4 a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {5 c (b d-2 a e) x^4+5 b^2 d-18 a c d-a b e}{x^2 \left (c x^8+b x^4+a\right )}dx}{4 a \left (b^2-4 a c\right )}+\frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 1828

\(\displaystyle \frac {-\frac {\int \frac {x^2 \left (c \left (5 d b^2-a e b-18 a c d\right ) x^4+5 b^3 d-23 a b c d-a b^2 e+10 a^2 c e\right )}{c x^8+b x^4+a}dx}{a}-\frac {-a b e-18 a c d+5 b^2 d}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 1834

\(\displaystyle \frac {-\frac {\frac {1}{2} c \left (\frac {20 a^2 c e-a b^2 e-28 a b c d+5 b^3 d}{\sqrt {b^2-4 a c}}-a b e-18 a c d+5 b^2 d\right ) \int \frac {2 x^2}{2 c x^4+b-\sqrt {b^2-4 a c}}dx+\frac {1}{2} c \left (-\frac {20 a^2 c e-a b^2 e-28 a b c d+5 b^3 d}{\sqrt {b^2-4 a c}}-a b e-18 a c d+5 b^2 d\right ) \int \frac {2 x^2}{2 c x^4+b+\sqrt {b^2-4 a c}}dx}{a}-\frac {-a b e-18 a c d+5 b^2 d}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {c \left (\frac {20 a^2 c e-a b^2 e-28 a b c d+5 b^3 d}{\sqrt {b^2-4 a c}}-a b e-18 a c d+5 b^2 d\right ) \int \frac {x^2}{2 c x^4+b-\sqrt {b^2-4 a c}}dx+c \left (-\frac {20 a^2 c e-a b^2 e-28 a b c d+5 b^3 d}{\sqrt {b^2-4 a c}}-a b e-18 a c d+5 b^2 d\right ) \int \frac {x^2}{2 c x^4+b+\sqrt {b^2-4 a c}}dx}{a}-\frac {-a b e-18 a c d+5 b^2 d}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {-\frac {c \left (-\frac {20 a^2 c e-a b^2 e-28 a b c d+5 b^3 d}{\sqrt {b^2-4 a c}}-a b e-18 a c d+5 b^2 d\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {-b-\sqrt {b^2-4 a c}}}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )+c \left (\frac {20 a^2 c e-a b^2 e-28 a b c d+5 b^3 d}{\sqrt {b^2-4 a c}}-a b e-18 a c d+5 b^2 d\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {\sqrt {b^2-4 a c}-b}}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )}{a}-\frac {-a b e-18 a c d+5 b^2 d}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {c \left (-\frac {20 a^2 c e-a b^2 e-28 a b c d+5 b^3 d}{\sqrt {b^2-4 a c}}-a b e-18 a c d+5 b^2 d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )+c \left (\frac {20 a^2 c e-a b^2 e-28 a b c d+5 b^3 d}{\sqrt {b^2-4 a c}}-a b e-18 a c d+5 b^2 d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )}{a}-\frac {-a b e-18 a c d+5 b^2 d}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {c \left (-\frac {20 a^2 c e-a b^2 e-28 a b c d+5 b^3 d}{\sqrt {b^2-4 a c}}-a b e-18 a c d+5 b^2 d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )+c \left (\frac {20 a^2 c e-a b^2 e-28 a b c d+5 b^3 d}{\sqrt {b^2-4 a c}}-a b e-18 a c d+5 b^2 d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{a}-\frac {-a b e-18 a c d+5 b^2 d}{a x}}{4 a \left (b^2-4 a c\right )}+\frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{4 a x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

Input:

Int[(d + e*x^4)/(x^2*(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)/(4*a*(b^2 - 4*a*c)*x*(a + 
b*x^4 + c*x^8)) + (-((5*b^2*d - 18*a*c*d - a*b*e)/(a*x)) - (c*(5*b^2*d - 1 
8*a*c*d - a*b*e - (5*b^3*d - 28*a*b*c*d - a*b^2*e + 20*a^2*c*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*2^ 
(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - ArcTanh[(2^(1/4)*c^(1/4)*x 
)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4*a* 
c])^(1/4))) + c*(5*b^2*d - 18*a*c*d - a*b*e + (5*b^3*d - 28*a*b*c*d - a*b^ 
2*e + 20*a^2*c*e)/Sqrt[b^2 - 4*a*c])*(ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + Sqr 
t[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) 
- ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c 
^(3/4)*(-b + Sqrt[b^2 - 4*a*c])^(1/4))))/a)/(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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 827

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

rule 1824

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

rule 1828

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

rule 1834

Int[(((f_.)*(x_))^(m_.)*((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[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 
 - (2*c*d - b*e)/(2*q))   Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ 
[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n 
, 0]
 

Maple [C] (verified)

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

Time = 0.17 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.32

method	result	size
default	\(\frac {\frac {-\frac {c \left (a b e +2 a c d -d \,b^{2}\right ) x^{7}}{4 \left (4 a c -b^{2}\right )}+\frac {\left (2 a^{2} c e -a \,b^{2} e -3 a b c d +b^{3} d \right ) x^{3}}{16 a c -4 b^{2}}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (-a b e -18 a c d +5 d \,b^{2}\right ) \textit {\_R}^{6}+\left (10 a^{2} c e -a \,b^{2} e -23 a b c d +5 b^{3} d \right ) \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a c -16 b^{2}}}{a^{2}}-\frac {d}{a^{2} x}\)	\(214\)
risch	\(\text {Expression too large to display}\)	\(4783\)

Input:

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

Output:

1/a^2*((-1/4*c*(a*b*e+2*a*c*d-b^2*d)/(4*a*c-b^2)*x^7+1/4*(2*a^2*c*e-a*b^2* 
e-3*a*b*c*d+b^3*d)/(4*a*c-b^2)*x^3)/(c*x^8+b*x^4+a)+1/16/(4*a*c-b^2)*sum(( 
c*(-a*b*e-18*a*c*d+5*b^2*d)*_R^6+(10*a^2*c*e-a*b^2*e-23*a*b*c*d+5*b^3*d)*_ 
R^2)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*c+_Z^4*b+a)))-d/a^2/x
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/4*((a*b*c*e - (5*b^2*c - 18*a*c^2)*d)*x^8 - ((5*b^3 - 19*a*b*c)*d - (a*b 
^2 - 2*a^2*c)*e)*x^4 - 4*(a*b^2 - 4*a^2*c)*d)/((a^2*b^2*c - 4*a^3*c^2)*x^9 
 + (a^2*b^3 - 4*a^3*b*c)*x^5 + (a^3*b^2 - 4*a^4*c)*x) + 1/4*integrate(((a* 
b*c*e - (5*b^2*c - 18*a*c^2)*d)*x^6 - ((5*b^3 - 23*a*b*c)*d - (a*b^2 - 10* 
a^2*c)*e)*x^2)/(c*x^8 + b*x^4 + a), x)/(a^2*b^2 - 4*a^3*c)
 

Giac [F]

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

integrate((e*x^4+d)/x^2/(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^2), x)
 

Mupad [B] (verification not implemented)

Time = 32.99 (sec) , antiderivative size = 142799, normalized size of antiderivative = 216.69 \[ \int \frac {d+e x^4}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((-(625*b^25*d^4 + a^4*b^21*e^4 + 625*b^10*d^4*(-(4*a*c - b^2)^15)^( 
1/2) + 3105423360*a^12*b*c^12*d^4 - 69*a^5*b^19*c*e^4 + 73728000*a^14*b*c^ 
10*e^4 - 20*a^3*b^22*d*e^3 - 1911029760*a^13*c^12*d^3*e + 589824000*a^14*c 
^11*d*e^3 + 638475*a^2*b^21*c^2*d^4 - 8264990*a^3*b^19*c^3*d^4 + 71483001* 
a^4*b^17*c^4*d^4 - 434478624*a^5*b^15*c^5*d^4 + 1898983360*a^6*b^13*c^6*d^ 
4 - 5996689920*a^7*b^11*c^7*d^4 + 13524825600*a^8*b^9*c^8*d^4 - 2112231014 
4*a^9*b^7*c^9*d^4 + 21483012096*a^10*b^5*c^10*d^4 - 12575047680*a^11*b^3*c 
^11*d^4 + a^4*b^6*e^4*(-(4*a*c - b^2)^15)^(1/2) - 26244*a^5*c^5*d^4*(-(4*a 
*c - b^2)^15)^(1/2) + 2085*a^6*b^17*c^2*e^4 - 36320*a^7*b^15*c^3*e^4 + 404 
160*a^8*b^13*c^4*e^4 - 3001344*a^9*b^11*c^5*e^4 + 15064576*a^10*b^9*c^6*e^ 
4 - 50503680*a^11*b^7*c^7*e^4 + 108380160*a^12*b^5*c^8*e^4 - 134676480*a^1 
3*b^3*c^9*e^4 - 2500*a^7*c^3*e^4*(-(4*a*c - b^2)^15)^(1/2) + 150*a^2*b^23* 
d^2*e^2 - 29625*a*b^23*c*d^4 - 500*a*b^24*d^3*e + 68475*a^2*b^6*c^2*d^4*(- 
(4*a*c - b^2)^15)^(1/2) - 181990*a^3*b^4*c^3*d^4*(-(4*a*c - b^2)^15)^(1/2) 
 + 171801*a^4*b^2*c^4*d^4*(-(4*a*c - b^2)^15)^(1/2) + 525*a^6*b^2*c^2*e^4* 
(-(4*a*c - b^2)^15)^(1/2) + 150*a^2*b^8*d^2*e^2*(-(4*a*c - b^2)^15)^(1/2) 
+ 224244*a^4*b^19*c^2*d^2*e^2 - 3363546*a^5*b^17*c^3*d^2*e^2 + 32811840*a^ 
6*b^15*c^4*d^2*e^2 - 219012480*a^7*b^13*c^5*d^2*e^2 + 1022161920*a^8*b^11* 
c^6*d^2*e^2 - 3338689536*a^9*b^9*c^7*d^2*e^2 + 7481769984*a^10*b^7*c^8*d^2 
*e^2 - 10951557120*a^11*b^5*c^9*d^2*e^2 + 9413591040*a^12*b^3*c^10*d^2*...
 

Reduce [F]

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

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

Output:

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