\(\int \frac {1}{x^5 (d+e x^4) (a+b x^4+c x^8)^2} \, dx\) [109]

Optimal result

Integrand size = 27, antiderivative size = 507 \[ \int \frac {1}{x^5 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=-\frac {1}{4 a^2 d x^4}-\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e+c \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) x^4}{4 a^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^4+c x^8\right )}+\frac {c \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )}-\frac {\left (2 b^2 c^2 d^3+2 b^4 d e^2-2 a c^2 d \left (c d^2+2 a e^2\right )+a b c e \left (5 c d^2+6 a e^2\right )-b^3 \left (4 c d^2 e+3 a e^3\right )\right ) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a^3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}-\frac {(2 b d+a e) \log (x)}{a^3 d^2}+\frac {e^5 \log \left (d+e x^4\right )}{4 d^2 \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (2 b^3 d e^2+2 b c d \left (c d^2+a e^2\right )+a c e \left (c d^2+2 a e^2\right )-b^2 \left (4 c d^2 e+3 a e^3\right )\right ) \log \left (a+b x^4+c x^8\right )}{8 a^3 \left (c d^2-b d e+a e^2\right )^2} \] Output:

-1/4/a^2/d/x^4-1/4*(b^3*c*d-3*a*b*c^2*d-b^4*e+4*a*b^2*c*e-2*a^2*c^2*e+c*(3 
*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)*x^4)/a^2/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2 
)/(c*x^8+b*x^4+a)+1/2*c*(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)*arctanh((2*c*x 
^4+b)/(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)-1/4*( 
2*b^2*c^2*d^3+2*b^4*d*e^2-2*a*c^2*d*(2*a*e^2+c*d^2)+a*b*c*e*(6*a*e^2+5*c*d 
^2)-b^3*(3*a*e^3+4*c*d^2*e))*arctanh((2*c*x^4+b)/(-4*a*c+b^2)^(1/2))/a^3/( 
-4*a*c+b^2)^(1/2)/(a*e^2-b*d*e+c*d^2)^2-(a*e+2*b*d)*ln(x)/a^3/d^2+1/4*e^5* 
ln(e*x^4+d)/d^2/(a*e^2-b*d*e+c*d^2)^2+1/8*(2*b^3*d*e^2+2*b*c*d*(a*e^2+c*d^ 
2)+a*c*e*(2*a*e^2+c*d^2)-b^2*(3*a*e^3+4*c*d^2*e))*ln(c*x^8+b*x^4+a)/a^3/(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 = 1.13 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x^5 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {1}{4} \left (-\frac {1}{a^2 d x^4}+\frac {-b^4 e-2 a c^2 \left (a e+c d x^4\right )+b^2 c \left (4 a e+c d x^4\right )+b^3 c \left (d-e x^4\right )-3 a b c^2 \left (d-e x^4\right )}{a^2 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \left (a+b x^4+c x^8\right )}-\frac {4 (2 b d+a e) \log (x)}{a^3 d^2}+\frac {e^5 \log \left (d+e x^4\right )}{\left (c d^3+d e (-b d+a e)\right )^2}-\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {2 b^4 c^2 d^3 \log (x-\text {$\#$1})-10 a b^2 c^3 d^3 \log (x-\text {$\#$1})+6 a^2 c^4 d^3 \log (x-\text {$\#$1})-4 b^5 c d^2 e \log (x-\text {$\#$1})+21 a b^3 c^2 d^2 e \log (x-\text {$\#$1})-17 a^2 b c^3 d^2 e \log (x-\text {$\#$1})+2 b^6 d e^2 \log (x-\text {$\#$1})-8 a b^4 c d e^2 \log (x-\text {$\#$1})-4 a^2 b^2 c^2 d e^2 \log (x-\text {$\#$1})+10 a^3 c^3 d e^2 \log (x-\text {$\#$1})-3 a b^5 e^3 \log (x-\text {$\#$1})+17 a^2 b^3 c e^3 \log (x-\text {$\#$1})-19 a^3 b c^2 e^3 \log (x-\text {$\#$1})+2 b^3 c^3 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^4-8 a b c^4 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^4-4 b^4 c^2 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+17 a b^2 c^3 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4-4 a^2 c^4 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+2 b^5 c d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-6 a b^3 c^2 d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-8 a^2 b c^3 d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-3 a b^4 c e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4+14 a^2 b^2 c^2 e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4-8 a^3 c^3 e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{b+2 c \text {$\#$1}^4}\&\right ]}{a^3 \left (-b^2+4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2}\right ) \] Input:

Integrate[1/(x^5*(d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

(-(1/(a^2*d*x^4)) + (-(b^4*e) - 2*a*c^2*(a*e + c*d*x^4) + b^2*c*(4*a*e + c 
*d*x^4) + b^3*c*(d - e*x^4) - 3*a*b*c^2*(d - e*x^4))/(a^2*(b^2 - 4*a*c)*(- 
(c*d^2) + e*(b*d - a*e))*(a + b*x^4 + c*x^8)) - (4*(2*b*d + a*e)*Log[x])/( 
a^3*d^2) + (e^5*Log[d + e*x^4])/(c*d^3 + d*e*(-(b*d) + a*e))^2 - RootSum[a 
 + b*#1^4 + c*#1^8 & , (2*b^4*c^2*d^3*Log[x - #1] - 10*a*b^2*c^3*d^3*Log[x 
 - #1] + 6*a^2*c^4*d^3*Log[x - #1] - 4*b^5*c*d^2*e*Log[x - #1] + 21*a*b^3* 
c^2*d^2*e*Log[x - #1] - 17*a^2*b*c^3*d^2*e*Log[x - #1] + 2*b^6*d*e^2*Log[x 
 - #1] - 8*a*b^4*c*d*e^2*Log[x - #1] - 4*a^2*b^2*c^2*d*e^2*Log[x - #1] + 1 
0*a^3*c^3*d*e^2*Log[x - #1] - 3*a*b^5*e^3*Log[x - #1] + 17*a^2*b^3*c*e^3*L 
og[x - #1] - 19*a^3*b*c^2*e^3*Log[x - #1] + 2*b^3*c^3*d^3*Log[x - #1]*#1^4 
 - 8*a*b*c^4*d^3*Log[x - #1]*#1^4 - 4*b^4*c^2*d^2*e*Log[x - #1]*#1^4 + 17* 
a*b^2*c^3*d^2*e*Log[x - #1]*#1^4 - 4*a^2*c^4*d^2*e*Log[x - #1]*#1^4 + 2*b^ 
5*c*d*e^2*Log[x - #1]*#1^4 - 6*a*b^3*c^2*d*e^2*Log[x - #1]*#1^4 - 8*a^2*b* 
c^3*d*e^2*Log[x - #1]*#1^4 - 3*a*b^4*c*e^3*Log[x - #1]*#1^4 + 14*a^2*b^2*c 
^2*e^3*Log[x - #1]*#1^4 - 8*a^3*c^3*e^3*Log[x - #1]*#1^4)/(b + 2*c*#1^4) & 
 ]/(a^3*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2))/4
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 502, 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.

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

\(\Big \downarrow \) 1802

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

\(\Big \downarrow \) 1289

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \left (-\frac {\text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right ) \left (-b^3 \left (3 a e^3+4 c d^2 e\right )+a b c e \left (6 a e^2+5 c d^2\right )-2 a c^2 d \left (2 a e^2+c d^2\right )+2 b^4 d e^2+2 b^2 c^2 d^3\right )}{a^3 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}+\frac {\left (-b^2 \left (3 a e^3+4 c d^2 e\right )+2 b c d \left (a e^2+c d^2\right )+a c e \left (2 a e^2+c d^2\right )+2 b^3 d e^2\right ) \log \left (a+b x^4+c x^8\right )}{2 a^3 \left (a e^2-b d e+c d^2\right )^2}-\frac {\log \left (x^4\right ) (a e+2 b d)}{a^3 d^2}+\frac {2 c \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right ) \left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right )}{a^2 \left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-2 a^2 c^2 e+4 a b^2 c e+c x^4 \left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right )-3 a b c^2 d+b^4 (-e)+b^3 c d}{a^2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right ) \left (a e^2-b d e+c d^2\right )}-\frac {1}{a^2 d x^4}+\frac {e^5 \log \left (d+e x^4\right )}{d^2 \left (a e^2-b d e+c d^2\right )^2}\right )\)

Input:

Int[1/(x^5*(d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

(-(1/(a^2*d*x^4)) - (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c 
^2*e + c*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)*x^4)/(a^2*(b^2 - 4*a*c) 
*(c*d^2 - b*d*e + a*e^2)*(a + b*x^4 + c*x^8)) + (2*c*(b^2*c*d - 2*a*c^2*d 
- b^3*e + 3*a*b*c*e)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(a^2*(b^2 - 
 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)) - ((2*b^2*c^2*d^3 + 2*b^4*d*e^2 - 2 
*a*c^2*d*(c*d^2 + 2*a*e^2) + a*b*c*e*(5*c*d^2 + 6*a*e^2) - b^3*(4*c*d^2*e 
+ 3*a*e^3))*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(a^3*Sqrt[b^2 - 4*a* 
c]*(c*d^2 - b*d*e + a*e^2)^2) - ((2*b*d + a*e)*Log[x^4])/(a^3*d^2) + (e^5* 
Log[d + e*x^4])/(d^2*(c*d^2 - b*d*e + a*e^2)^2) + ((2*b^3*d*e^2 + 2*b*c*d* 
(c*d^2 + a*e^2) + a*c*e*(c*d^2 + 2*a*e^2) - b^2*(4*c*d^2*e + 3*a*e^3))*Log 
[a + b*x^4 + c*x^8])/(2*a^3*(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 = 52.36 (sec) , antiderivative size = 836, normalized size of antiderivative = 1.65

method	result	size
default	\(\frac {\frac {\frac {a c \left (3 a^{2} b c \,e^{3}-2 a^{2} c^{2} e^{2} d -a \,b^{3} e^{3}-2 a \,b^{2} c d \,e^{2}+5 a b \,c^{2} d^{2} e -2 a \,c^{3} d^{3}+b^{4} d \,e^{2}-2 b^{3} c \,d^{2} e +b^{2} c^{2} d^{3}\right ) x^{4}}{4 a c -b^{2}}-\frac {a \left (2 a^{3} c^{2} e^{3}-4 a^{2} b^{2} c \,e^{3}+a^{2} b \,c^{2} d \,e^{2}+2 a^{2} c^{3} d^{2} e +a \,b^{4} e^{3}+3 a \,b^{3} c d \,e^{2}-7 a \,b^{2} c^{2} d^{2} e +3 a b \,c^{3} d^{3}-b^{5} d \,e^{2}+2 b^{4} c \,d^{2} e -b^{3} c^{2} d^{3}\right )}{4 a c -b^{2}}}{2 c \,x^{8}+2 b \,x^{4}+2 a}+\frac {\frac {\left (8 a^{3} c^{3} e^{3}-14 a^{2} b^{2} c^{2} e^{3}+8 a^{2} b \,c^{3} d \,e^{2}+4 a^{2} c^{4} d^{2} e +3 a \,b^{4} c \,e^{3}+6 a \,b^{3} c^{2} d \,e^{2}-17 a \,b^{2} c^{3} d^{2} e +8 a b \,c^{4} d^{3}-2 b^{5} c d \,e^{2}+4 b^{4} c^{2} d^{2} e -2 d^{3} c^{3} b^{3}\right ) \ln \left (c \,x^{8}+b \,x^{4}+a \right )}{2 c}+\frac {2 \left (19 a^{3} b \,c^{2} e^{3}-10 a^{3} c^{3} d \,e^{2}-17 a^{2} b^{3} c \,e^{3}+4 a^{2} b^{2} c^{2} d \,e^{2}+17 a^{2} b \,c^{3} d^{2} e -6 a^{2} c^{4} d^{3}+3 a \,b^{5} e^{3}+8 a \,b^{4} c d \,e^{2}-21 a \,b^{3} c^{2} d^{2} e +10 a \,b^{2} c^{3} d^{3}-2 b^{6} d \,e^{2}+4 b^{5} c \,d^{2} e -2 b^{4} c^{2} d^{3}-\frac {\left (8 a^{3} c^{3} e^{3}-14 a^{2} b^{2} c^{2} e^{3}+8 a^{2} b \,c^{3} d \,e^{2}+4 a^{2} c^{4} d^{2} e +3 a \,b^{4} c \,e^{3}+6 a \,b^{3} c^{2} d \,e^{2}-17 a \,b^{2} c^{3} d^{2} e +8 a b \,c^{4} d^{3}-2 b^{5} c d \,e^{2}+4 b^{4} c^{2} d^{2} e -2 d^{3} c^{3} b^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{8 a c -2 b^{2}}}{2 a^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {e^{5} \ln \left (x^{4} e +d \right )}{4 d^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {1}{4 a^{2} d \,x^{4}}+\frac {\left (-a e -2 b d \right ) \ln \left (x \right )}{a^{3} d^{2}}\)	\(836\)
risch	\(\text {Expression too large to display}\)	\(4844\)

Input:

int(1/x^5/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2/a^3/(a*e^2-b*d*e+c*d^2)^2*(1/2*(a*c*(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)*x^4-a*(2*a^3*c^2*e^3-4*a^2*b^2*c*e^3+a^2*b*c^2*d 
*e^2+2*a^2*c^3*d^2*e+a*b^4*e^3+3*a*b^3*c*d*e^2-7*a*b^2*c^2*d^2*e+3*a*b*c^3 
*d^3-b^5*d*e^2+2*b^4*c*d^2*e-b^3*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^3*c^3*e^3-14*a^2*b^2*c^2*e^3+8*a^2*b*c^3*d*e^2+4*a^ 
2*c^4*d^2*e+3*a*b^4*c*e^3+6*a*b^3*c^2*d*e^2-17*a*b^2*c^3*d^2*e+8*a*b*c^4*d 
^3-2*b^5*c*d*e^2+4*b^4*c^2*d^2*e-2*b^3*c^3*d^3)/c*ln(c*x^8+b*x^4+a)+2*(19* 
a^3*b*c^2*e^3-10*a^3*c^3*d*e^2-17*a^2*b^3*c*e^3+4*a^2*b^2*c^2*d*e^2+17*a^2 
*b*c^3*d^2*e-6*a^2*c^4*d^3+3*a*b^5*e^3+8*a*b^4*c*d*e^2-21*a*b^3*c^2*d^2*e+ 
10*a*b^2*c^3*d^3-2*b^6*d*e^2+4*b^5*c*d^2*e-2*b^4*c^2*d^3-1/2*(8*a^3*c^3*e^ 
3-14*a^2*b^2*c^2*e^3+8*a^2*b*c^3*d*e^2+4*a^2*c^4*d^2*e+3*a*b^4*c*e^3+6*a*b 
^3*c^2*d*e^2-17*a*b^2*c^3*d^2*e+8*a*b*c^4*d^3-2*b^5*c*d*e^2+4*b^4*c^2*d^2* 
e-2*b^3*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^5*ln(e*x^4+d)/d^2/(a*e^2-b*d*e+c*d^2)^2-1/4/a^2/d/x^4+(-a*e-2* 
b*d)/a^3/d^2*ln(x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/x^5/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^5 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x^5/(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. 1303 vs. \(2 (487) = 974\).

Time = 4.44 (sec) , antiderivative size = 1303, normalized size of antiderivative = 2.57 \[ \int \frac {1}{x^5 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/x^5/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
 

Output:

1/4*e^6*log(abs(e*x^4 + d))/(c^2*d^6*e - 2*b*c*d^5*e^2 + b^2*d^4*e^3 + 2*a 
*c*d^4*e^3 - 2*a*b*d^3*e^4 + a^2*d^2*e^5) + 1/8*(2*b*c^2*d^3 - 4*b^2*c*d^2 
*e + a*c^2*d^2*e + 2*b^3*d*e^2 + 2*a*b*c*d*e^2 - 3*a*b^2*e^3 + 2*a^2*c*e^3 
)*log(c*x^8 + b*x^4 + a)/(a^3*c^2*d^4 - 2*a^3*b*c*d^3*e + a^3*b^2*d^2*e^2 
+ 2*a^4*c*d^2*e^2 - 2*a^4*b*d*e^3 + a^5*e^4) + 1/4*(2*b^4*c^2*d^3 - 12*a*b 
^2*c^3*d^3 + 12*a^2*c^4*d^3 - 4*b^5*c*d^2*e + 25*a*b^3*c^2*d^2*e - 30*a^2* 
b*c^3*d^2*e + 2*b^6*d*e^2 - 10*a*b^4*c*d*e^2 + 20*a^3*c^3*d*e^2 - 3*a*b^5* 
e^3 + 20*a^2*b^3*c*e^3 - 30*a^3*b*c^2*e^3)*arctan((2*c*x^4 + b)/sqrt(-b^2 
+ 4*a*c))/((a^3*b^2*c^2*d^4 - 4*a^4*c^3*d^4 - 2*a^3*b^3*c*d^3*e + 8*a^4*b* 
c^2*d^3*e + a^3*b^4*d^2*e^2 - 2*a^4*b^2*c*d^2*e^2 - 8*a^5*c^2*d^2*e^2 - 2* 
a^4*b^3*d*e^3 + 8*a^5*b*c*d*e^3 + a^5*b^2*e^4 - 4*a^6*c*e^4)*sqrt(-b^2 + 4 
*a*c)) + 1/12*(a^2*b^2*c*e^5*x^12 - 4*a^3*c^2*e^5*x^12 - 6*b^2*c^3*d^5*x^8 
 + 18*a*c^4*d^5*x^8 + 12*b^3*c^2*d^4*e*x^8 - 39*a*b*c^3*d^4*e*x^8 - 6*b^4* 
c*d^3*e^2*x^8 + 12*a*b^2*c^2*d^3*e^2*x^8 + 30*a^2*c^3*d^3*e^2*x^8 + 9*a*b^ 
3*c*d^2*e^3*x^8 - 33*a^2*b*c^2*d^2*e^3*x^8 - 3*a^2*b^2*c*d*e^4*x^8 + 12*a^ 
3*c^2*d*e^4*x^8 + a^2*b^3*e^5*x^8 - 4*a^3*b*c*e^5*x^8 - 6*b^3*c^2*d^5*x^4 
+ 21*a*b*c^3*d^5*x^4 + 12*b^4*c*d^4*e*x^4 - 45*a*b^2*c^2*d^4*e*x^4 + 6*a^2 
*c^3*d^4*e*x^4 - 6*b^5*d^3*e^2*x^4 + 15*a*b^3*c*d^3*e^2*x^4 + 27*a^2*b*c^2 
*d^3*e^2*x^4 + 9*a*b^4*d^2*e^3*x^4 - 36*a^2*b^2*c*d^2*e^3*x^4 + 6*a^3*c^2* 
d^2*e^3*x^4 - 3*a^2*b^3*d*e^4*x^4 + 12*a^3*b*c*d*e^4*x^4 + a^3*b^2*e^5*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Hanged} \] Input:

int(1/(x^5*(d + e*x^4)*(a + b*x^4 + c*x^8)^2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1}{x^5 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {1}{x^{5} \left (e \,x^{4}+d \right ) \left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:

int(1/x^5/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
 

Output:

int(1/x^5/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)