\(\int \frac {1}{x^2 (d+e x^4) (a+b x^4+c x^8)} \, dx\) [87]

Optimal result

Integrand size = 27, antiderivative size = 709 \[ \int \frac {1}{x^2 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=-\frac {1}{a d x}-\frac {\sqrt [4]{c} \left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} a \sqrt [4]{-b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt [4]{c} \left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} a \sqrt [4]{-b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}+\frac {e^{9/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{2 \sqrt {2} d^{5/4} \left (c d^2-b d e+a e^2\right )}-\frac {e^{9/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{2 \sqrt {2} d^{5/4} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt [4]{c} \left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} a \sqrt [4]{-b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt [4]{c} \left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} a \sqrt [4]{-b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}+\frac {e^{9/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x}{\sqrt {d}+\sqrt {e} x^2}\right )}{2 \sqrt {2} d^{5/4} \left (c d^2-b d e+a e^2\right )} \] Output:

-1/a/d/x-1/4*c^(1/4)*(c*d-b*e-(2*a*c*e-b^2*e+b*c*d)/(-4*a*c+b^2)^(1/2))*ar 
ctan(2^(1/4)*c^(1/4)*x/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(1/4)/a/(-b-(-4*a* 
c+b^2)^(1/2))^(1/4)/(a*e^2-b*d*e+c*d^2)-1/4*c^(1/4)*(c*d-b*e+(2*a*c*e-b^2* 
e+b*c*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/(-b+(-4*a*c+b^2)^(1/2))^(1/4)/(a*e^2-b*d*e+c*d^2)-1/4 
*e^(9/4)*arctan(-1+2^(1/2)*e^(1/4)*x/d^(1/4))*2^(1/2)/d^(5/4)/(a*e^2-b*d*e 
+c*d^2)-1/4*e^(9/4)*arctan(1+2^(1/2)*e^(1/4)*x/d^(1/4))*2^(1/2)/d^(5/4)/(a 
*e^2-b*d*e+c*d^2)+1/4*c^(1/4)*(c*d-b*e-(2*a*c*e-b^2*e+b*c*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/ 
(-b-(-4*a*c+b^2)^(1/2))^(1/4)/(a*e^2-b*d*e+c*d^2)+1/4*c^(1/4)*(c*d-b*e+(2* 
a*c*e-b^2*e+b*c*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/(-b+(-4*a*c+b^2)^(1/2))^(1/4)/(a*e^2-b*d*e 
+c*d^2)+1/4*e^(9/4)*arctanh(2^(1/2)*d^(1/4)*e^(1/4)*x/(d^(1/2)+e^(1/2)*x^2 
))*2^(1/2)/d^(5/4)/(a*e^2-b*d*e+c*d^2)
 

Mathematica [C] (verified)

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

Time = 0.31 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^2 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\frac {-8 c d^{9/4}+8 b d^{5/4} e-8 a \sqrt [4]{d} e^2+2 \sqrt {2} a e^{9/4} x \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )-2 \sqrt {2} a e^{9/4} x \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )-\sqrt {2} a e^{9/4} x \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {e} x^2\right )+\sqrt {2} a e^{9/4} x \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {e} x^2\right )+2 d^{5/4} x \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b c d \log (x-\text {$\#$1})+b^2 e \log (x-\text {$\#$1})-a c e \log (x-\text {$\#$1})-c^2 d \log (x-\text {$\#$1}) \text {$\#$1}^4+b c e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{8 a d^{5/4} \left (c d^2+e (-b d+a e)\right ) x} \] Input:

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

Output:

(-8*c*d^(9/4) + 8*b*d^(5/4)*e - 8*a*d^(1/4)*e^2 + 2*Sqrt[2]*a*e^(9/4)*x*Ar 
cTan[1 - (Sqrt[2]*e^(1/4)*x)/d^(1/4)] - 2*Sqrt[2]*a*e^(9/4)*x*ArcTan[1 + ( 
Sqrt[2]*e^(1/4)*x)/d^(1/4)] - Sqrt[2]*a*e^(9/4)*x*Log[Sqrt[d] - Sqrt[2]*d^ 
(1/4)*e^(1/4)*x + Sqrt[e]*x^2] + Sqrt[2]*a*e^(9/4)*x*Log[Sqrt[d] + Sqrt[2] 
*d^(1/4)*e^(1/4)*x + Sqrt[e]*x^2] + 2*d^(5/4)*x*RootSum[a + b*#1^4 + c*#1^ 
8 & , (-(b*c*d*Log[x - #1]) + b^2*e*Log[x - #1] - a*c*e*Log[x - #1] - c^2* 
d*Log[x - #1]*#1^4 + b*c*e*Log[x - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(8*a*d 
^(5/4)*(c*d^2 + e*(-(b*d) + a*e))*x)
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 778, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1836, 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.

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

\(\Big \downarrow \) 1836

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt [4]{c} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right ) \left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right )}{2\ 2^{3/4} a \sqrt [4]{-\sqrt {b^2-4 a c}-b} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt [4]{c} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right ) \left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt {b^2-4 a c}-b} \left (a e^2-b d e+c d^2\right )}+\frac {e^{9/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{2 \sqrt {2} d^{5/4} \left (a e^2-b d e+c d^2\right )}-\frac {e^{9/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}+1\right )}{2 \sqrt {2} d^{5/4} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right ) \left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right )}{2\ 2^{3/4} a \sqrt [4]{-\sqrt {b^2-4 a c}-b} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right ) \left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt {b^2-4 a c}-b} \left (a e^2-b d e+c d^2\right )}-\frac {e^{9/4} \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {d}+\sqrt {e} x^2\right )}{4 \sqrt {2} d^{5/4} \left (a e^2-b d e+c d^2\right )}+\frac {e^{9/4} \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {d}+\sqrt {e} x^2\right )}{4 \sqrt {2} d^{5/4} \left (a e^2-b d e+c d^2\right )}-\frac {1}{a d x}\)

Input:

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

Output:

-(1/(a*d*x)) - (c^(1/4)*(c*d - b*e - (b*c*d - b^2*e + 2*a*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)*a*(-b - Sqrt[b^2 - 4*a*c])^(1/4)*(c*d^2 - b*d*e + a*e^2)) - (c^(1/4)* 
(c*d - b*e + (b*c*d - b^2*e + 2*a*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)*a*(-b + Sqrt[b^2 - 
4*a*c])^(1/4)*(c*d^2 - b*d*e + a*e^2)) + (e^(9/4)*ArcTan[1 - (Sqrt[2]*e^(1 
/4)*x)/d^(1/4)])/(2*Sqrt[2]*d^(5/4)*(c*d^2 - b*d*e + a*e^2)) - (e^(9/4)*Ar 
cTan[1 + (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(2*Sqrt[2]*d^(5/4)*(c*d^2 - b*d*e + 
 a*e^2)) + (c^(1/4)*(c*d - b*e - (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a* 
c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4 
)*a*(-b - Sqrt[b^2 - 4*a*c])^(1/4)*(c*d^2 - b*d*e + a*e^2)) + (c^(1/4)*(c* 
d - b*e + (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^ 
(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b + Sqrt[b^2 - 4* 
a*c])^(1/4)*(c*d^2 - b*d*e + a*e^2)) - (e^(9/4)*Log[Sqrt[d] - Sqrt[2]*d^(1 
/4)*e^(1/4)*x + Sqrt[e]*x^2])/(4*Sqrt[2]*d^(5/4)*(c*d^2 - b*d*e + a*e^2)) 
+ (e^(9/4)*Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*x + Sqrt[e]*x^2])/(4*Sqrt 
[2]*d^(5/4)*(c*d^2 - b*d*e + a*e^2))
 

Defintions of rubi rules used

rule 1836

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.))/((a_) + (c_.)*(x_)^ 
(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e 
*x^n)^q/(a + b*x^n + c*x^(2*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] && IntegerQ[q] && Int 
egerQ[m]
 

rule 2009

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

Maple [C] (verified)

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

Time = 0.42 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.32

method	result	size
default	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (-e b +c d \right ) \textit {\_R}^{6}+\left (a c e -b^{2} e +c b d \right ) \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) a}-\frac {1}{a d x}-\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{e}}}{x^{2}+\left (\frac {d}{e}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{e}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{e}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{e}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\frac {d}{e}\right )^{\frac {1}{4}}}\)	\(228\)
risch	\(\text {Expression too large to display}\)	\(16659\)

Input:

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

Output:

-1/4/(a*e^2-b*d*e+c*d^2)/a*sum((c*(-b*e+c*d)*_R^6+(a*c*e-b^2*e+b*c*d)*_R^2 
)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))-1/a/d/x-1/8*e^2/d 
/(a*e^2-b*d*e+c*d^2)/(d/e)^(1/4)*2^(1/2)*(ln((x^2-(d/e)^(1/4)*x*2^(1/2)+(d 
/e)^(1/2))/(x^2+(d/e)^(1/4)*x*2^(1/2)+(d/e)^(1/2)))+2*arctan(2^(1/2)/(d/e) 
^(1/4)*x+1)+2*arctan(2^(1/2)/(d/e)^(1/4)*x-1))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/8*e^3*(sqrt(2)*log(sqrt(e)*x^2 + sqrt(2)*d^(1/4)*e^(1/4)*x + sqrt(d))/(d 
^(1/4)*e^(3/4)) - sqrt(2)*log(sqrt(e)*x^2 - sqrt(2)*d^(1/4)*e^(1/4)*x + sq 
rt(d))/(d^(1/4)*e^(3/4)) - sqrt(2)*log((2*sqrt(e)*x - sqrt(2)*sqrt(-sqrt(d 
)*sqrt(e)) + sqrt(2)*d^(1/4)*e^(1/4))/(2*sqrt(e)*x + sqrt(2)*sqrt(-sqrt(d) 
*sqrt(e)) + sqrt(2)*d^(1/4)*e^(1/4)))/(sqrt(-sqrt(d)*sqrt(e))*sqrt(e)) - s 
qrt(2)*log((2*sqrt(e)*x - sqrt(2)*sqrt(-sqrt(d)*sqrt(e)) - sqrt(2)*d^(1/4) 
*e^(1/4))/(2*sqrt(e)*x + sqrt(2)*sqrt(-sqrt(d)*sqrt(e)) - sqrt(2)*d^(1/4)* 
e^(1/4)))/(sqrt(-sqrt(d)*sqrt(e))*sqrt(e)))/(c*d^3 - b*d^2*e + a*d*e^2) + 
integrate(-((c^2*d - b*c*e)*x^6 + (b*c*d - (b^2 - a*c)*e)*x^2)/(c*x^8 + b* 
x^4 + a), x)/(a*c*d^2 - a*b*d*e + a^2*e^2) - 1/(a*d*x)
 

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 39.25 (sec) , antiderivative size = 276728, normalized size of antiderivative = 390.31 \[ \int \frac {1}{x^2 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((-e^9/(256*c^4*d^13 + 256*a^4*d^5*e^8 + 256*b^4*d^9*e^4 - 1024*a*b^3 
*d^8*e^5 - 1024*a^3*b*d^6*e^7 + 1024*a*c^3*d^11*e^2 + 1024*a^3*c*d^7*e^6 - 
 1024*b^3*c*d^10*e^3 + 1536*a^2*b^2*d^7*e^6 + 1536*a^2*c^2*d^9*e^4 + 1536* 
b^2*c^2*d^11*e^2 - 1024*b*c^3*d^12*e - 3072*a*b*c^2*d^10*e^3 + 3072*a*b^2* 
c*d^9*e^4 - 3072*a^2*b*c*d^8*e^5))^(1/4)*(x*(4*a^19*c^13*d^20*e^12 + 4*a^1 
9*b*c^12*d^19*e^13) + (-e^9/(256*c^4*d^13 + 256*a^4*d^5*e^8 + 256*b^4*d^9* 
e^4 - 1024*a*b^3*d^8*e^5 - 1024*a^3*b*d^6*e^7 + 1024*a*c^3*d^11*e^2 + 1024 
*a^3*c*d^7*e^6 - 1024*b^3*c*d^10*e^3 + 1536*a^2*b^2*d^7*e^6 + 1536*a^2*c^2 
*d^9*e^4 + 1536*b^2*c^2*d^11*e^2 - 1024*b*c^3*d^12*e - 3072*a*b*c^2*d^10*e 
^3 + 3072*a*b^2*c*d^9*e^4 - 3072*a^2*b*c*d^8*e^5))^(3/4)*((x*(163840*a^24* 
c^12*d^23*e^13 - 327680*a^25*c^11*d^21*e^15 + 32768*a^26*c^10*d^19*e^17 - 
4096*a^19*b^2*c^15*d^31*e^5 + 6144*a^19*b^3*c^14*d^30*e^6 - 4096*a^19*b^4* 
c^13*d^29*e^7 + 1024*a^19*b^5*c^12*d^28*e^8 + 1024*a^19*b^9*c^8*d^24*e^12 
- 4096*a^19*b^10*c^7*d^23*e^13 + 6144*a^19*b^11*c^6*d^22*e^14 - 4096*a^19* 
b^12*c^5*d^21*e^15 + 1024*a^19*b^13*c^4*d^20*e^16 - 12288*a^20*b^2*c^14*d^ 
29*e^7 + 12288*a^20*b^3*c^13*d^28*e^8 - 4096*a^20*b^4*c^12*d^27*e^9 - 1331 
2*a^20*b^7*c^9*d^24*e^12 + 58368*a^20*b^8*c^8*d^23*e^13 - 96256*a^20*b^9*c 
^7*d^22*e^14 + 71680*a^20*b^10*c^6*d^21*e^15 - 21504*a^20*b^11*c^5*d^20*e^ 
16 + 1024*a^20*b^12*c^4*d^19*e^17 - 12288*a^21*b^2*c^13*d^27*e^9 + 6144*a^ 
21*b^3*c^12*d^26*e^10 + 62464*a^21*b^5*c^10*d^24*e^12 - 311296*a^21*b^6...
 

Reduce [F]

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

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

Output:

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