\(\int \frac {1}{x^3 (d+e x^4) (a+b x^4+c x^8)^2} \, dx\) [114]

Optimal result

Integrand size = 27, antiderivative size = 928 \[ \int \frac {1}{x^3 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=-\frac {1}{2 a^2 d x^2}-\frac {x^2 \left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e+c \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) x^4\right )}{4 a^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^4+c x^8\right )}-\frac {\sqrt {c} \left (3 b^5 d e^2-b^4 e \left (6 c d^2-e \left (3 \sqrt {b^2-4 a c} d-5 a e\right )\right )-a b c \left (16 c^2 d^3-19 a \sqrt {b^2-4 a c} e^3-3 c d e \left (7 \sqrt {b^2-4 a c} d-4 a e\right )\right )+b^3 \left (3 c^2 d^3-5 a \sqrt {b^2-4 a c} e^3-6 c d e \left (\sqrt {b^2-4 a c} d+2 a e\right )\right )-b^2 c \left (a e^2 \left (6 \sqrt {b^2-4 a c} d-29 a e\right )-3 c d^2 \left (\sqrt {b^2-4 a c} d+11 a e\right )\right )-2 a c^2 \left (c d^2 \left (5 \sqrt {b^2-4 a c} d+6 a e\right )+a e^2 \left (9 \sqrt {b^2-4 a c} d+14 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-e (b d-a e)\right )^2}+\frac {\sqrt {c} \left (3 b^5 d e^2+2 a c^2 \left (a e^2 \left (9 \sqrt {b^2-4 a c} d-14 a e\right )+c d^2 \left (5 \sqrt {b^2-4 a c} d-6 a e\right )\right )+b^3 \left (3 c^2 d^3+5 a \sqrt {b^2-4 a c} e^3+6 c d e \left (\sqrt {b^2-4 a c} d-2 a e\right )\right )-a b c \left (16 c^2 d^3+19 a \sqrt {b^2-4 a c} e^3+3 c d e \left (7 \sqrt {b^2-4 a c} d+4 a e\right )\right )-b^4 e \left (6 c d^2+e \left (3 \sqrt {b^2-4 a c} d+5 a e\right )\right )-b^2 c \left (3 c d^2 \left (\sqrt {b^2-4 a c} d-11 a e\right )-a e^2 \left (6 \sqrt {b^2-4 a c} d+29 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-e (b d-a e)\right )^2}-\frac {e^{9/2} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2-b d e+a e^2\right )^2} \] Output:

-1/2/a^2/d/x^2-1/4*x^2*(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)-1/8*c^(1/2)*(3*b^5*d*e^2-b^4*e*(6*c*d^2-e*(3*(-4*a*c 
+b^2)^(1/2)*d-5*a*e))-a*b*c*(16*c^2*d^3-19*a*(-4*a*c+b^2)^(1/2)*e^3-3*c*d* 
e*(7*(-4*a*c+b^2)^(1/2)*d-4*a*e))+b^3*(3*c^2*d^3-5*a*(-4*a*c+b^2)^(1/2)*e^ 
3-6*c*d*e*((-4*a*c+b^2)^(1/2)*d+2*a*e))-b^2*c*(a*e^2*(6*(-4*a*c+b^2)^(1/2) 
*d-29*a*e)-3*c*d^2*((-4*a*c+b^2)^(1/2)*d+11*a*e))-2*a*c^2*(c*d^2*(5*(-4*a* 
c+b^2)^(1/2)*d+6*a*e)+a*e^2*(9*(-4*a*c+b^2)^(1/2)*d+14*a*e)))*arctan(2^(1/ 
2)*c^(1/2)*x^2/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^2/(-4*a*c+b^2)^(3/2 
)/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(c*d^2-e*(-a*e+b*d))^2+1/8*c^(1/2)*(3*b^5*d 
*e^2+2*a*c^2*(a*e^2*(9*(-4*a*c+b^2)^(1/2)*d-14*a*e)+c*d^2*(5*(-4*a*c+b^2)^ 
(1/2)*d-6*a*e))+b^3*(3*c^2*d^3+5*a*(-4*a*c+b^2)^(1/2)*e^3+6*c*d*e*((-4*a*c 
+b^2)^(1/2)*d-2*a*e))-a*b*c*(16*c^2*d^3+19*a*(-4*a*c+b^2)^(1/2)*e^3+3*c*d* 
e*(7*(-4*a*c+b^2)^(1/2)*d+4*a*e))-b^4*e*(6*c*d^2+e*(3*(-4*a*c+b^2)^(1/2)*d 
+5*a*e))-b^2*c*(3*c*d^2*((-4*a*c+b^2)^(1/2)*d-11*a*e)-a*e^2*(6*(-4*a*c+b^2 
)^(1/2)*d+29*a*e)))*arctan(2^(1/2)*c^(1/2)*x^2/(b+(-4*a*c+b^2)^(1/2))^(1/2 
))*2^(1/2)/a^2/(-4*a*c+b^2)^(3/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(c*d^2-e*(- 
a*e+b*d))^2-1/2*e^(9/2)*arctan(e^(1/2)*x^2/d^(1/2))/d^(3/2)/(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 = 1.16 (sec) , antiderivative size = 702, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^3 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {1}{8} \left (-\frac {4}{a^2 d x^2}-\frac {2 x^2 \left (b^4 e+2 a c^2 \left (a e+c d x^4\right )-b^2 c \left (4 a e+c d x^4\right )+3 a b c^2 \left (d-e x^4\right )+b^3 c \left (-d+e x^4\right )\right )}{a^2 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \left (a+b x^4+c x^8\right )}+\frac {4 e^{9/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{d^{3/2} \left (c d^2+e (-b d+a e)\right )^2}+\frac {4 e^{9/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{d^{3/2} \left (c d^2+e (-b d+a e)\right )^2}+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {3 b^3 c^2 d^3 \log (x-\text {$\#$1})-13 a b c^3 d^3 \log (x-\text {$\#$1})-6 b^4 c d^2 e \log (x-\text {$\#$1})+27 a b^2 c^2 d^2 e \log (x-\text {$\#$1})-6 a^2 c^3 d^2 e \log (x-\text {$\#$1})+3 b^5 d e^2 \log (x-\text {$\#$1})-9 a b^3 c d e^2 \log (x-\text {$\#$1})-15 a^2 b c^2 d e^2 \log (x-\text {$\#$1})-5 a b^4 e^3 \log (x-\text {$\#$1})+24 a^2 b^2 c e^3 \log (x-\text {$\#$1})-14 a^3 c^2 e^3 \log (x-\text {$\#$1})+3 b^2 c^3 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^4-10 a c^4 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^4-6 b^3 c^2 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+21 a b c^3 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+3 b^4 c d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-6 a b^2 c^2 d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-18 a^2 c^3 d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-5 a b^3 c e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4+19 a^2 b c^2 e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}^2+2 c \text {$\#$1}^6}\&\right ]}{a^2 \left (-b^2+4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2}\right ) \] Input:

Integrate[1/(x^3*(d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

(-4/(a^2*d*x^2) - (2*x^2*(b^4*e + 2*a*c^2*(a*e + c*d*x^4) - b^2*c*(4*a*e + 
 c*d*x^4) + 3*a*b*c^2*(d - e*x^4) + b^3*c*(-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*e^(9/2)*ArcTan[1 - 
(Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(d^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^2) + (4* 
e^(9/2)*ArcTan[1 + (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(d^(3/2)*(c*d^2 + e*(-(b* 
d) + a*e))^2) + RootSum[a + b*#1^4 + c*#1^8 & , (3*b^3*c^2*d^3*Log[x - #1] 
 - 13*a*b*c^3*d^3*Log[x - #1] - 6*b^4*c*d^2*e*Log[x - #1] + 27*a*b^2*c^2*d 
^2*e*Log[x - #1] - 6*a^2*c^3*d^2*e*Log[x - #1] + 3*b^5*d*e^2*Log[x - #1] - 
 9*a*b^3*c*d*e^2*Log[x - #1] - 15*a^2*b*c^2*d*e^2*Log[x - #1] - 5*a*b^4*e^ 
3*Log[x - #1] + 24*a^2*b^2*c*e^3*Log[x - #1] - 14*a^3*c^2*e^3*Log[x - #1] 
+ 3*b^2*c^3*d^3*Log[x - #1]*#1^4 - 10*a*c^4*d^3*Log[x - #1]*#1^4 - 6*b^3*c 
^2*d^2*e*Log[x - #1]*#1^4 + 21*a*b*c^3*d^2*e*Log[x - #1]*#1^4 + 3*b^4*c*d* 
e^2*Log[x - #1]*#1^4 - 6*a*b^2*c^2*d*e^2*Log[x - #1]*#1^4 - 18*a^2*c^3*d*e 
^2*Log[x - #1]*#1^4 - 5*a*b^3*c*e^3*Log[x - #1]*#1^4 + 19*a^2*b*c^2*e^3*Lo 
g[x - #1]*#1^4)/(b*#1^2 + 2*c*#1^6) & ]/(a^2*(-b^2 + 4*a*c)*(c*d^2 + e*(-( 
b*d) + a*e))^2))/8
 

Rubi [A] (verified)

Time = 4.15 (sec) , antiderivative size = 888, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1814, 1674, 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.

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

\(\Big \downarrow \) 1814

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

\(\Big \downarrow \) 1674

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {\arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right ) e^{9/2}}{d^{3/2} \left (c d^2-b e d+a e^2\right )^2}-\frac {\sqrt {c} \left (-e b^3+c d b^2+3 a c e b-2 a c^2 d+\frac {-e b^4+c d b^3+9 a c e b^2-8 a c^2 d b-12 a^2 c^2 e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )}-\frac {\sqrt {c} \left ((c d-b e) \left (c d^2-e (b d-2 a e)\right )+\frac {d e^2 b^3-2 \left (a e^3+c d^2 e\right ) b^2+c d \left (c d^2+2 a e^2\right ) b+2 a^2 c e^3}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )^2}-\frac {\sqrt {c} \left (-e b^3+c d b^2+3 a c e b-2 a c^2 d-\frac {-e b^4+c d b^3+9 a c e b^2-8 a c^2 d b-12 a^2 c^2 e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )}-\frac {\sqrt {c} \left ((c d-b e) \left (c d^2-e (b d-2 a e)\right )-\frac {d e^2 b^3-2 \left (a e^3+c d^2 e\right ) b^2+c d \left (c d^2+2 a e^2\right ) b+2 a^2 c e^3}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )^2}-\frac {x^2 \left (-e b^4+c d b^3+4 a c e b^2-3 a c^2 d b+c \left (-e b^3+c d b^2+3 a c e b-2 a c^2 d\right ) x^4-2 a^2 c^2 e\right )}{2 a^2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}-\frac {1}{a^2 d x^2}\right )\)

Input:

Int[1/(x^3*(d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

(-(1/(a^2*d*x^2)) - (x^2*(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))/(2*a^2*(b^2 
- 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x^4 + c*x^8)) - (Sqrt[c]*(b^2*c*d 
- 2*a*c^2*d - b^3*e + 3*a*b*c*e + (b^3*c*d - 8*a*b*c^2*d - b^4*e + 9*a*b^2 
*c*e - 12*a^2*c^2*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[ 
b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^2*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 
4*a*c]]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[c]*((c*d - b*e)*(c*d^2 - e*(b*d - 
 2*a*e)) + (b^3*d*e^2 + 2*a^2*c*e^3 + b*c*d*(c*d^2 + 2*a*e^2) - 2*b^2*(c*d 
^2*e + a*e^3))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b - Sq 
rt[b^2 - 4*a*c]]])/(Sqrt[2]*a^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e 
 + a*e^2)^2) - (Sqrt[c]*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e - (b^3*c* 
d - 8*a*b*c^2*d - b^4*e + 9*a*b^2*c*e - 12*a^2*c^2*e)/Sqrt[b^2 - 4*a*c])*A 
rcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^2*( 
b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[ 
c]*((c*d - b*e)*(c*d^2 - e*(b*d - 2*a*e)) - (b^3*d*e^2 + 2*a^2*c*e^3 + b*c 
*d*(c*d^2 + 2*a*e^2) - 2*b^2*(c*d^2*e + a*e^3))/Sqrt[b^2 - 4*a*c])*ArcTan[ 
(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a^2*Sqrt[b + 
Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^2) - (e^(9/2)*ArcTan[(Sqrt[e]*x 
^2)/Sqrt[d]])/(d^(3/2)*(c*d^2 - b*d*e + a*e^2)^2))/2
 

Defintions of rubi rules used

rule 1674

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && N 
eQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0] || IntegersQ[m, q])
 

rule 1814

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e 
_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Sub 
st[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + b*x^(n/k) + c*x^(2*(n/k)))^ 
p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 
 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
 

rule 2009

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

Maple [A] (verified)

Time = 7.33 (sec) , antiderivative size = 1081, normalized size of antiderivative = 1.16

method	result	size
default	\(\text {Expression too large to display}\)	\(1081\)
risch	\(\text {Expression too large to display}\)	\(24170\)

Input:

int(1/x^3/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/2/(a*e^2-b*d*e+c*d^2)^2/a^2*((-1/2*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^6+1/2*(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)*x^2)/(c*x^8+b*x^4+ 
a)+2/(4*a*c-b^2)*c*(-1/8*(-19*a^2*b*c*e^3*(-4*a*c+b^2)^(1/2)+18*a^2*c^2*e^ 
2*d*(-4*a*c+b^2)^(1/2)+5*a*b^3*e^3*(-4*a*c+b^2)^(1/2)+6*a*b^2*c*d*e^2*(-4* 
a*c+b^2)^(1/2)-21*a*b*c^2*d^2*e*(-4*a*c+b^2)^(1/2)+10*a*c^3*d^3*(-4*a*c+b^ 
2)^(1/2)-3*b^4*d*e^2*(-4*a*c+b^2)^(1/2)+6*b^3*c*d^2*e*(-4*a*c+b^2)^(1/2)-3 
*b^2*c^2*d^3*(-4*a*c+b^2)^(1/2)+28*a^3*c^2*e^3-29*a^2*b^2*c*e^3+12*a^2*b*c 
^2*d*e^2+12*a^2*c^3*d^2*e+5*a*b^4*e^3+12*a*b^3*c*d*e^2-33*a*b^2*c^2*d^2*e+ 
16*a*b*c^3*d^3-3*b^5*d*e^2+6*b^4*c*d^2*e-3*b^3*c^2*d^3)/(-4*a*c+b^2)^(1/2) 
*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^2*2^(1/2)/((-b+(-4* 
a*c+b^2)^(1/2))*c)^(1/2))+1/8*(-19*a^2*b*c*e^3*(-4*a*c+b^2)^(1/2)+18*a^2*c 
^2*e^2*d*(-4*a*c+b^2)^(1/2)+5*a*b^3*e^3*(-4*a*c+b^2)^(1/2)+6*a*b^2*c*d*e^2 
*(-4*a*c+b^2)^(1/2)-21*a*b*c^2*d^2*e*(-4*a*c+b^2)^(1/2)+10*a*c^3*d^3*(-4*a 
*c+b^2)^(1/2)-3*b^4*d*e^2*(-4*a*c+b^2)^(1/2)+6*b^3*c*d^2*e*(-4*a*c+b^2)^(1 
/2)-3*b^2*c^2*d^3*(-4*a*c+b^2)^(1/2)-28*a^3*c^2*e^3+29*a^2*b^2*c*e^3-12*a^ 
2*b*c^2*d*e^2-12*a^2*c^3*d^2*e-5*a*b^4*e^3-12*a*b^3*c*d*e^2+33*a*b^2*c^2*d 
^2*e-16*a*b*c^3*d^3+3*b^5*d*e^2-6*b^4*c*d^2*e+3*b^3*c^2*d^3)/(-4*a*c+b^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/x^3/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x^3/(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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28419 vs. \(2 (836) = 1672\).

Time = 6.79 (sec) , antiderivative size = 28419, normalized size of antiderivative = 30.62 \[ \int \frac {1}{x^3 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/x^3/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
 

Output:

-1/2*e^5*arctan(e*x^2/sqrt(d*e))/((c^2*d^5 - 2*b*c*d^4*e + b^2*d^3*e^2 + 2 
*a*c*d^3*e^2 - 2*a*b*d^2*e^3 + a^2*d*e^4)*sqrt(d*e)) - 1/16*((6*a^2*b^5*c^ 
7 - 44*a^3*b^3*c^8 + 80*a^4*b*c^9 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + 
 sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^5 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
+ sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^6 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
+ sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^6 - 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
 + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^7 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
+ sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^7 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
+ sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^7 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
 + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^8 - 6*(b^2 - 4*a*c)*a^2*b^3*c^7 + 20*(b^2 
- 4*a*c)*a^3*b*c^8)*d^7*x^4 - (24*a^2*b^6*c^6 - 178*a^3*b^4*c^7 + 328*a^4* 
b^2*c^8 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2 
*b^6*c^4 + 89*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^ 
3*b^4*c^5 + 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a 
^2*b^5*c^5 - 164*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) 
*a^4*b^2*c^6 - 82*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c 
)*a^3*b^3*c^6 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)* 
c)*a^2*b^4*c^6 + 41*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c) 
*c)*a^3*b^2*c^7 - 24*(b^2 - 4*a*c)*a^2*b^4*c^6 + 82*(b^2 - 4*a*c)*a^3*b^2* 
c^7)*d^6*e*x^4 + 2*(18*a^2*b^7*c^5 - 124*a^3*b^5*c^6 + 170*a^4*b^3*c^7 ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Hanged} \] Input:

int(1/(x^3*(d + e*x^4)*(a + b*x^4 + c*x^8)^2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1}{x^3 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {1}{x^{3} \left (e \,x^{4}+d \right ) \left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:

int(1/x^3/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
 

Output:

int(1/x^3/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)