\(\int \frac {1}{x^2 (d+e x^4) (a+b x^4+c x^8)^2} \, dx\) [119]

Optimal result

Integrand size = 27, antiderivative size = 2110 \[ \int \frac {1}{x^2 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx =\text {Too large to display} \] Output:

-1/a^2/d/x+1/5*(5*b^3*d*e^2+a^2*c*e^3+5*b*c*d*(2*a*e^2+c*d^2)-10*b^2*(a*e^ 
3+c*d^2*e))*x^3/a^2/(a*e^2-b*d*e+c*d^2)^2/(c*x^8+b*x^4+a)+c*(-b*e+c*d)*(c* 
d^2-e*(-2*a*e+b*d))*x^7/a^2/(a*e^2-b*d*e+c*d^2)^2/(c*x^8+b*x^4+a)-1/20*x^3 
*(25*b^5*d*e^2+5*b^3*c*d*(-11*a*e^2+5*c*d^2)-2*a^2*c^2*e*(13*a*e^2+5*c*d^2 
)-5*a*b*c^2*d*(33*a*e^2+19*c*d^2)+a*b^2*c*e*(184*a*e^2+195*c*d^2)-5*b^4*(9 
*a*e^3+10*c*d^2*e)+5*c*(5*b^4*d*e^2+5*b^2*c*d*(-2*a*e^2+c*d^2)-2*a*c^2*d*( 
17*a*e^2+9*c*d^2)+a*b*c*e*(35*a*e^2+37*c*d^2)-b^3*(9*a*e^3+10*c*d^2*e))*x^ 
4)/a^2/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(c*x^8+b*x^4+a)+1/16*c^(1/4)*(5* 
b^5*d*e^2+2*a*c^2*(a*e^2*(17*(-4*a*c+b^2)^(1/2)*d-26*a*e)+c*d^2*(9*(-4*a*c 
+b^2)^(1/2)*d-10*a*e))+b^3*(5*c^2*d^3+9*a*(-4*a*c+b^2)^(1/2)*e^3+10*c*d*e* 
((-4*a*c+b^2)^(1/2)*d-2*a*e))-b^4*e*(10*c*d^2+e*(5*(-4*a*c+b^2)^(1/2)*d+9* 
a*e))-a*b*c*(28*c^2*d^3+35*a*(-4*a*c+b^2)^(1/2)*e^3+c*d*e*(37*(-4*a*c+b^2) 
^(1/2)*d+24*a*e))-b^2*c*(c*d^2*(5*(-4*a*c+b^2)^(1/2)*d-57*a*e)-a*e^2*(10*( 
-4*a*c+b^2)^(1/2)*d+53*a*e)))*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)^(3/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)/( 
a*e^2-b*d*e+c*d^2)^2-1/16*c^(1/4)*(5*b^5*d*e^2-a*b*c*(28*c^2*d^3-35*a*(-4* 
a*c+b^2)^(1/2)*e^3-c*d*e*(37*(-4*a*c+b^2)^(1/2)*d-24*a*e))-b^4*e*(10*c*d^2 
-e*(5*(-4*a*c+b^2)^(1/2)*d-9*a*e))+b^3*(5*c^2*d^3-9*a*(-4*a*c+b^2)^(1/2)*e 
^3-10*c*d*e*((-4*a*c+b^2)^(1/2)*d+2*a*e))-2*a*c^2*(c*d^2*(9*(-4*a*c+b^2)^( 
1/2)*d+10*a*e)+a*e^2*(17*(-4*a*c+b^2)^(1/2)*d+26*a*e))-b^2*c*(a*e^2*(10...
 

Mathematica [C] (verified)

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

Time = 10.69 (sec) , antiderivative size = 847, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x^2 \left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {1}{16} \left (-\frac {16}{a^2 d x}-\frac {4 x^3 \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 \sqrt {2} e^{17/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{d^{5/4} \left (c d^2+e (-b d+a e)\right )^2}-\frac {4 \sqrt {2} e^{17/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{d^{5/4} \left (c d^2+e (-b d+a e)\right )^2}-\frac {2 \sqrt {2} e^{17/4} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {e} x^2\right )}{d^{5/4} \left (c d^2+e (-b d+a e)\right )^2}+\frac {2 \sqrt {2} e^{17/4} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {e} x^2\right )}{d^{5/4} \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 {5 b^3 c^2 d^3 \log (x-\text {$\#$1})-23 a b c^3 d^3 \log (x-\text {$\#$1})-10 b^4 c d^2 e \log (x-\text {$\#$1})+47 a b^2 c^2 d^2 e \log (x-\text {$\#$1})-10 a^2 c^3 d^2 e \log (x-\text {$\#$1})+5 b^5 d e^2 \log (x-\text {$\#$1})-15 a b^3 c d e^2 \log (x-\text {$\#$1})-29 a^2 b c^2 d e^2 \log (x-\text {$\#$1})-9 a b^4 e^3 \log (x-\text {$\#$1})+44 a^2 b^2 c e^3 \log (x-\text {$\#$1})-26 a^3 c^2 e^3 \log (x-\text {$\#$1})+5 b^2 c^3 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^4-18 a c^4 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^4-10 b^3 c^2 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+37 a b c^3 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+5 b^4 c d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-10 a b^2 c^2 d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-34 a^2 c^3 d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-9 a b^3 c e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4+35 a^2 b c^2 e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\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^2*(d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

(-16/(a^2*d*x) - (4*x^3*(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*Sqrt[2]*e^(17/4)*Arc 
Tan[1 - (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(d^(5/4)*(c*d^2 + e*(-(b*d) + a*e))^ 
2) - (4*Sqrt[2]*e^(17/4)*ArcTan[1 + (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(d^(5/4) 
*(c*d^2 + e*(-(b*d) + a*e))^2) - (2*Sqrt[2]*e^(17/4)*Log[Sqrt[d] - Sqrt[2] 
*d^(1/4)*e^(1/4)*x + Sqrt[e]*x^2])/(d^(5/4)*(c*d^2 + e*(-(b*d) + a*e))^2) 
+ (2*Sqrt[2]*e^(17/4)*Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*x + Sqrt[e]*x^ 
2])/(d^(5/4)*(c*d^2 + e*(-(b*d) + a*e))^2) + RootSum[a + b*#1^4 + c*#1^8 & 
 , (5*b^3*c^2*d^3*Log[x - #1] - 23*a*b*c^3*d^3*Log[x - #1] - 10*b^4*c*d^2* 
e*Log[x - #1] + 47*a*b^2*c^2*d^2*e*Log[x - #1] - 10*a^2*c^3*d^2*e*Log[x - 
#1] + 5*b^5*d*e^2*Log[x - #1] - 15*a*b^3*c*d*e^2*Log[x - #1] - 29*a^2*b*c^ 
2*d*e^2*Log[x - #1] - 9*a*b^4*e^3*Log[x - #1] + 44*a^2*b^2*c*e^3*Log[x - # 
1] - 26*a^3*c^2*e^3*Log[x - #1] + 5*b^2*c^3*d^3*Log[x - #1]*#1^4 - 18*a*c^ 
4*d^3*Log[x - #1]*#1^4 - 10*b^3*c^2*d^2*e*Log[x - #1]*#1^4 + 37*a*b*c^3*d^ 
2*e*Log[x - #1]*#1^4 + 5*b^4*c*d*e^2*Log[x - #1]*#1^4 - 10*a*b^2*c^2*d*e^2 
*Log[x - #1]*#1^4 - 34*a^2*c^3*d*e^2*Log[x - #1]*#1^4 - 9*a*b^3*c*e^3*Log[ 
x - #1]*#1^4 + 35*a^2*b*c^2*e^3*Log[x - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(a 
^2*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2))/16
 

Rubi [F]

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.

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

\(\Big \downarrow \) 1887

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 1887

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( 
(d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Unintegrable[(f*x)^m*(d + e*x^n 
)^q*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, 
x] && EqQ[n2, 2*n]
 

Maple [C] (verified)

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

Time = 8.98 (sec) , antiderivative size = 697, normalized size of antiderivative = 0.33

method	result	size
default	\(-\frac {\frac {-\frac {c \left (3 a^{2} b c \,e^{3}-2 a^{2} c^{2} e^{2} d -a \,b^{3} e^{3}-2 a \,b^{2} c d \,e^{2}+5 a b \,c^{2} d^{2} e -2 a \,c^{3} d^{3}+b^{4} d \,e^{2}-2 b^{3} c \,d^{2} e +b^{2} c^{2} d^{3}\right ) x^{7}}{4 \left (4 a c -b^{2}\right )}+\frac {\left (2 a^{3} c^{2} e^{3}-4 a^{2} b^{2} c \,e^{3}+a^{2} b \,c^{2} d \,e^{2}+2 a^{2} c^{3} d^{2} e +a \,b^{4} e^{3}+3 a \,b^{3} c d \,e^{2}-7 a \,b^{2} c^{2} d^{2} e +3 a b \,c^{3} d^{3}-b^{5} d \,e^{2}+2 b^{4} c \,d^{2} e -b^{3} c^{2} d^{3}\right ) x^{3}}{16 a c -4 b^{2}}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (-35 a^{2} b c \,e^{3}+34 a^{2} c^{2} e^{2} d +9 a \,b^{3} e^{3}+10 a \,b^{2} c d \,e^{2}-37 a b \,c^{2} d^{2} e +18 a \,c^{3} d^{3}-5 b^{4} d \,e^{2}+10 b^{3} c \,d^{2} e -5 b^{2} c^{2} d^{3}\right ) \textit {\_R}^{6}+\left (26 a^{3} c^{2} e^{3}-44 a^{2} b^{2} c \,e^{3}+29 a^{2} b \,c^{2} d \,e^{2}+10 a^{2} c^{3} d^{2} e +9 a \,b^{4} e^{3}+15 a \,b^{3} c d \,e^{2}-47 a \,b^{2} c^{2} d^{2} e +23 a b \,c^{3} d^{3}-5 b^{5} d \,e^{2}+10 b^{4} c \,d^{2} e -5 b^{3} c^{2} d^{3}\right ) \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a c -16 b^{2}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} a^{2}}-\frac {e^{4} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{e}}}{x^{2}+\left (\frac {d}{e}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{e}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{e}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{e}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (\frac {d}{e}\right )^{\frac {1}{4}}}-\frac {1}{a^{2} d x}\)	\(697\)
risch	\(\text {Expression too large to display}\)	\(97672\)

Input:

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

Output:

-1/(a*e^2-b*d*e+c*d^2)^2/a^2*((-1/4*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^7+1/4*(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^3)/(c*x^8+b*x^4+a) 
+1/16/(4*a*c-b^2)*sum((c*(-35*a^2*b*c*e^3+34*a^2*c^2*d*e^2+9*a*b^3*e^3+10* 
a*b^2*c*d*e^2-37*a*b*c^2*d^2*e+18*a*c^3*d^3-5*b^4*d*e^2+10*b^3*c*d^2*e-5*b 
^2*c^2*d^3)*_R^6+(26*a^3*c^2*e^3-44*a^2*b^2*c*e^3+29*a^2*b*c^2*d*e^2+10*a^ 
2*c^3*d^2*e+9*a*b^4*e^3+15*a*b^3*c*d*e^2-47*a*b^2*c^2*d^2*e+23*a*b*c^3*d^3 
-5*b^5*d*e^2+10*b^4*c*d^2*e-5*b^3*c^2*d^3)*_R^2)/(2*_R^7*c+_R^3*b)*ln(x-_R 
),_R=RootOf(_Z^8*c+_Z^4*b+a)))-1/8*e^4/d/(a*e^2-b*d*e+c*d^2)^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))-1/a^2/d/x
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

1/8*e^5*(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^2*d^5 - 2*b*c*d^4*e - 2*a*b 
*d^2*e^3 + a^2*d*e^4 + (b^2 + 2*a*c)*d^3*e^2) - 1/4*(((5*b^2*c^2 - 18*a*c^ 
3)*d^2 - (5*b^3*c - 19*a*b*c^2)*d*e + 4*(a*b^2*c - 4*a^2*c^2)*e^2)*x^8 + ( 
(5*b^3*c - 19*a*b*c^2)*d^2 - (5*b^4 - 20*a*b^2*c + 2*a^2*c^2)*d*e + 4*(a*b 
^3 - 4*a^2*b*c)*e^2)*x^4 + 4*(a*b^2*c - 4*a^2*c^2)*d^2 - 4*(a*b^3 - 4*a^2* 
b*c)*d*e + 4*(a^2*b^2 - 4*a^3*c)*e^2)/(((a^2*b^2*c^2 - 4*a^3*c^3)*d^3 - (a 
^2*b^3*c - 4*a^3*b*c^2)*d^2*e + (a^3*b^2*c - 4*a^4*c^2)*d*e^2)*x^9 + ((a^2 
*b^3*c - 4*a^3*b*c^2)*d^3 - (a^2*b^4 - 4*a^3*b^2*c)*d^2*e + (a^3*b^3 - 4*a 
^4*b*c)*d*e^2)*x^5 + ((a^3*b^2*c - 4*a^4*c^2)*d^3 - (a^3*b^3 - 4*a^4*b*c)* 
d^2*e + (a^4*b^2 - 4*a^5*c)*d*e^2)*x) - 1/4*integrate((((5*b^2*c^3 - 18*a* 
c^4)*d^3 - (10*b^3*c^2 - 37*a*b*c^3)*d^2*e + (5*b^4*c - 10*a*b^2*c^2 - 34* 
a^2*c^3)*d*e^2 - (9*a*b^3*c - 35*a^2*b*c^2)*e^3)*x^6 + ((5*b^3*c^2 - 23*a* 
b*c^3)*d^3 - (10*b^4*c - 47*a*b^2*c^2 + 10*a^2*c^3)*d^2*e + (5*b^5 - 15...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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