\(\int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}+\frac {b}{x^4}} \, dx\) [59]

Optimal result

Integrand size = 22, antiderivative size = 433 \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}+\frac {b}{x^4}} \, dx=\frac {d x}{c}+\frac {\left (b d-c e+\frac {b^2 d-2 a c d-b c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}+\frac {\left (b d-c e-\frac {b^2 d-2 a c d-b c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}+\frac {\left (b d-c e+\frac {b^2 d-2 a c d-b c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}+\frac {\left (b d-c e-\frac {b^2 d-2 a c d-b c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}} \] Output:

d*x/c+1/4*(b*d-c*e+(-2*a*c*d+b^2*d-b*c*e)/(-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^(3/4)/c^(5/4)/(-b-(-4*a*c+b^ 
2)^(1/2))^(3/4)+1/4*(b*d-c*e-(-2*a*c*d+b^2*d-b*c*e)/(-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^(3/4)/c^(5/4)/(-b+ 
(-4*a*c+b^2)^(1/2))^(3/4)+1/4*(b*d-c*e+(-2*a*c*d+b^2*d-b*c*e)/(-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^(3/4)/c 
^(5/4)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)+1/4*(b*d-c*e-(-2*a*c*d+b^2*d-b*c*e)/( 
-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^(3/4)/c^(5/4)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.20 \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}+\frac {b}{x^4}} \, dx=\frac {d x}{c}-\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {a d \log (x-\text {$\#$1})+b d \log (x-\text {$\#$1}) \text {$\#$1}^4-c e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{4 c} \] Input:

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

Output:

(d*x)/c - RootSum[a + b*#1^4 + c*#1^8 & , (a*d*Log[x - #1] + b*d*Log[x - # 
1]*#1^4 - c*e*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]/(4*c)
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1727, 1826, 1752, 756, 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.

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

\(\Big \downarrow \) 1727

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

\(\Big \downarrow \) 1826

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

\(\Big \downarrow \) 1752

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(d*x)/c - (((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*(-(A 
rcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4) 
*(-b - Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt 
[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4))))/2 
 + ((b*d - c*e - (b^2*d - 2*a*c*d - b*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^(1/4)*c^(1/4)*(-b + S 
qrt[b^2 - 4*a*c])^(3/4))) - ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4 
*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))))/2)/c
 

Defintions of rubi rules used

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 756

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

rule 1727

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

rule 1752

Int[((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[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   I 
nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 
, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 
 - 4*a*c] ||  !IGtQ[n/2, 0])
 

rule 1826

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( 
c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a 
+ b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*( 
m + n*(2*p + 1) + 1))   Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a* 
e*(m - n + 1) + (b*e*(m + n*p + 1) - c*d*(m + n*(2*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] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && Intege 
rQ[p]
 

Maple [C] (verified)

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

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.15

method	result	size
default	\(\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (-b d +c e \right ) \textit {\_R}^{4}-a d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{4 c}\)	\(67\)
risch	\(\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (-b d +c e \right ) \textit {\_R}^{4}-a d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{4 c}\)	\(67\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 12946 vs. \(2 (353) = 706\).

Time = 4.09 (sec) , antiderivative size = 12946, normalized size of antiderivative = 29.90 \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}+\frac {b}{x^4}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 28.07 (sec) , antiderivative size = 50213, normalized size of antiderivative = 115.97 \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}+\frac {b}{x^4}} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((((4*x*(4096*a^5*b*c^6*d^2 + 4096*a^4*b*c^7*e^2 + 256*a^3*b^5*c^4*d 
^2 - 2048*a^4*b^3*c^5*d^2 + 256*a^2*b^5*c^5*e^2 - 2048*a^3*b^3*c^6*e^2 - 1 
6384*a^5*c^7*d*e - 1024*a^3*b^4*c^5*d*e + 8192*a^4*b^2*c^6*d*e))/c - (16*( 
-(b^9*d^4 + b^4*d^4*(-(4*a*c - b^2)^5)^(1/2) + b^5*c^4*e^4 + c^4*e^4*(-(4* 
a*c - b^2)^5)^(1/2) + 80*a^4*b*c^4*d^4 - 8*a*b^3*c^5*e^4 + 16*a^2*b*c^6*e^ 
4 + 128*a^3*c^6*d*e^3 - 128*a^4*c^5*d^3*e - 4*b^6*c^3*d*e^3 + 61*a^2*b^5*c 
^2*d^4 - 120*a^3*b^3*c^3*d^4 + a^2*c^2*d^4*(-(4*a*c - b^2)^5)^(1/2) + 6*b^ 
7*c^2*d^2*e^2 - 13*a*b^7*c*d^4 - 4*b^8*c*d^3*e + 240*a^2*b^3*c^4*d^2*e^2 + 
 6*b^2*c^2*d^2*e^2*(-(4*a*c - b^2)^5)^(1/2) - 3*a*b^2*c*d^4*(-(4*a*c - b^2 
)^5)^(1/2) + 40*a*b^4*c^4*d*e^3 + 48*a*b^6*c^2*d^3*e - 4*b*c^3*d*e^3*(-(4* 
a*c - b^2)^5)^(1/2) - 4*b^3*c*d^3*e*(-(4*a*c - b^2)^5)^(1/2) - 66*a*b^5*c^ 
3*d^2*e^2 - 128*a^2*b^2*c^5*d*e^3 - 200*a^2*b^4*c^3*d^3*e - 288*a^3*b*c^5* 
d^2*e^2 + 320*a^3*b^2*c^4*d^3*e - 6*a*c^3*d^2*e^2*(-(4*a*c - b^2)^5)^(1/2) 
 + 8*a*b*c^2*d^3*e*(-(4*a*c - b^2)^5)^(1/2))/(512*(256*a^4*c^9 + b^8*c^5 - 
 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8)))^(1/4)*(16384*a^5*c^8*e 
 - 256*a^2*b^6*c^5*e + 3072*a^3*b^4*c^6*e - 12288*a^4*b^2*c^7*e))/c)*(-(b^ 
9*d^4 + b^4*d^4*(-(4*a*c - b^2)^5)^(1/2) + b^5*c^4*e^4 + c^4*e^4*(-(4*a*c 
- b^2)^5)^(1/2) + 80*a^4*b*c^4*d^4 - 8*a*b^3*c^5*e^4 + 16*a^2*b*c^6*e^4 + 
128*a^3*c^6*d*e^3 - 128*a^4*c^5*d^3*e - 4*b^6*c^3*d*e^3 + 61*a^2*b^5*c^2*d 
^4 - 120*a^3*b^3*c^3*d^4 + a^2*c^2*d^4*(-(4*a*c - b^2)^5)^(1/2) + 6*b^7...
 

Reduce [F]

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

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

Output:

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