\(\int \frac {x^6 (d+e x^4)}{(a+b x^4+c x^8)^2} \, dx\) [100]

Optimal result

Integrand size = 25, antiderivative size = 527 \[ \int \frac {x^6 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=-\frac {x^3 \left (b d-2 a e+(2 c d-b e) x^4\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}-\frac {\left (2 c d-b e+\frac {8 b c d-b^2 e-12 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right ) \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\left (2 c d-b e-\frac {8 b c d-b^2 e-12 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt {b^2-4 a c}}}+\frac {\left (2 c d-b e+\frac {8 b c d-b^2 e-12 a c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right ) \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\left (2 c d-b e-\frac {8 b c d-b^2 e-12 a c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \] Output:

-1/4*x^3*(b*d-2*a*e+(-b*e+2*c*d)*x^4)/(-4*a*c+b^2)/(c*x^8+b*x^4+a)-1/16*(2 
*c*d-b*e+(-12*a*c*e-b^2*e+8*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)/c^(3/4)/(-4*a*c+b^2)/(-b-(-4* 
a*c+b^2)^(1/2))^(1/4)-1/16*(2*c*d-b*e-(-12*a*c*e-b^2*e+8*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)/ 
c^(3/4)/(-4*a*c+b^2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)+1/16*(2*c*d-b*e+(-12*a* 
c*e-b^2*e+8*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)/c^(3/4)/(-4*a*c+b^2)/(-b-(-4*a*c+b^2)^(1/2)) 
^(1/4)+1/16*(2*c*d-b*e-(-12*a*c*e-b^2*e+8*b*c*d)/(-4*a*c+b^2)^(1/2))*arcta 
nh(2^(1/4)*c^(1/4)*x/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*2^(1/4)/c^(3/4)/(-4*a* 
c+b^2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)
 

Mathematica [C] (verified)

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

Time = 0.24 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.27 \[ \int \frac {x^6 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {\frac {4 x^3 \left (-b d+2 a e-2 c d x^4+b e x^4\right )}{a+b x^4+c x^8}+\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {3 b d \log (x-\text {$\#$1})-6 a e \log (x-\text {$\#$1})-2 c d \log (x-\text {$\#$1}) \text {$\#$1}^4+b e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{16 \left (b^2-4 a c\right )} \] Input:

Integrate[(x^6*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x]
 

Output:

((4*x^3*(-(b*d) + 2*a*e - 2*c*d*x^4 + b*e*x^4))/(a + b*x^4 + c*x^8) + Root 
Sum[a + b*#1^4 + c*#1^8 & , (3*b*d*Log[x - #1] - 6*a*e*Log[x - #1] - 2*c*d 
*Log[x - #1]*#1^4 + b*e*Log[x - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(16*(b^2 
- 4*a*c))
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.81, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1822, 25, 1834, 27, 827, 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.

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

\(\Big \downarrow \) 1822

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1834

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

Int[(x^6*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x]
 

Output:

-1/4*(x^3*(b*d - 2*a*e + (2*c*d - b*e)*x^4))/((b^2 - 4*a*c)*(a + b*x^4 + c 
*x^8)) + (-((2*c*d - b*e + (8*b*c*d - b^2*e - 12*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)*c^ 
(3/4)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - 
Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4*a*c])^(1/4 
)))) - (2*c*d - b*e - (8*b*c*d - b^2*e - 12*a*c*e)/Sqrt[b^2 - 4*a*c])*(Arc 
Tan[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4) 
*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) - ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[ 
b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b + Sqrt[b^2 - 4*a*c])^(1/4))))/ 
(4*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 827

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

rule 1822

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( 
c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Simp[f^(n - 1)*(f*x)^(m - n + 1)*(a + 
b*x^n + c*x^(2*n))^(p + 1)*((b*d - 2*a*e - (b*e - 2*c*d)*x^n)/(n*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[f^n/(n*(p + 1)*(b^2 - 4*a*c))   Int[(f*x)^(m - n)* 
(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[(n - m - 1)*(b*d - 2*a*e) + (2*n*p + 2 
*n + m + 1)*(b*e - 2*c*d)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] & 
& EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, 
 n - 1] && IntegerQ[p]
 

rule 1834

Int[(((f_.)*(x_))^(m_.)*((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[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 
 - (2*c*d - b*e)/(2*q))   Int[(f*x)^m/(b/2 + q/2 + c*x^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]
 

Maple [C] (verified)

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

Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.28

method	result	size
default	\(\frac {-\frac {\left (e b -2 c d \right ) x^{7}}{4 \left (4 a c -b^{2}\right )}-\frac {\left (2 a e -b d \right ) x^{3}}{4 \left (4 a c -b^{2}\right )}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (-e b +2 c d \right ) \textit {\_R}^{6}+3 \left (2 a e -b d \right ) \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a c -16 b^{2}}\)	\(148\)
risch	\(\frac {-\frac {\left (e b -2 c d \right ) x^{7}}{4 \left (4 a c -b^{2}\right )}-\frac {\left (2 a e -b d \right ) x^{3}}{4 \left (4 a c -b^{2}\right )}}{c \,x^{8}+b \,x^{4}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-\frac {\left (e b -2 c d \right ) \textit {\_R}^{6}}{4 a c -b^{2}}+\frac {3 \left (2 a e -b d \right ) \textit {\_R}^{2}}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{16}\)	\(160\)

Input:

int(x^6*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

(-1/4*(b*e-2*c*d)/(4*a*c-b^2)*x^7-1/4*(2*a*e-b*d)/(4*a*c-b^2)*x^3)/(c*x^8+ 
b*x^4+a)+1/16/(4*a*c-b^2)*sum(((-b*e+2*c*d)*_R^6+3*(2*a*e-b*d)*_R^2)/(2*_R 
^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^6 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x^6*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^6 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**6*(e*x**4+d)/(c*x**8+b*x**4+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^6 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\int { \frac {{\left (e x^{4} + d\right )} x^{6}}{{\left (c x^{8} + b x^{4} + a\right )}^{2}} \,d x } \] Input:

integrate(x^6*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="maxima")
 

Output:

-1/4*((2*c*d - b*e)*x^7 + (b*d - 2*a*e)*x^3)/((b^2*c - 4*a*c^2)*x^8 + (b^3 
 - 4*a*b*c)*x^4 + a*b^2 - 4*a^2*c) + 1/4*integrate(-((2*c*d - b*e)*x^6 - 3 
*(b*d - 2*a*e)*x^2)/(c*x^8 + b*x^4 + a), x)/(b^2 - 4*a*c)
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^6 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x^6*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 28.44 (sec) , antiderivative size = 84889, normalized size of antiderivative = 161.08 \[ \int \frac {x^6 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

int((x^6*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x)
 

Output:

- atan(((((1769472*a*b^16*c^4*d^3 - 2147483648*a^9*c^12*d^3 - 86973087744* 
a^10*b*c^10*e^3 + 57982058496*a^10*c^11*d*e^2 - 38928384*a^2*b^14*c^5*d^3 
+ 339214336*a^3*b^12*c^6*d^3 - 1402994688*a^4*b^10*c^7*d^3 + 2139095040*a^ 
5*b^8*c^8*d^3 + 3388997632*a^6*b^6*c^9*d^3 - 16508780544*a^7*b^4*c^10*d^3 
+ 17716740096*a^8*b^2*c^11*d^3 + 65536*a^3*b^15*c^3*e^3 - 22806528*a^4*b^1 
3*c^4*e^3 + 525336576*a^5*b^11*c^5*e^3 - 5179965440*a^6*b^9*c^6*e^3 + 2743 
0748160*a^7*b^7*c^7*e^3 - 81939922944*a^8*b^5*c^8*e^3 + 130728067072*a^9*b 
^3*c^9*e^3 - 54760833024*a^9*b*c^11*d^2*e - 12386304*a^2*b^15*c^4*d^2*e + 
283901952*a^3*b^13*c^5*d^2*e + 27918336*a^3*b^14*c^4*d*e^2 - 2651848704*a^ 
4*b^11*c^6*d^2*e - 655884288*a^4*b^12*c^5*d*e^2 + 12645826560*a^5*b^9*c^7* 
d^2*e + 6360662016*a^5*b^10*c^6*d*e^2 - 30450647040*a^6*b^7*c^8*d^2*e - 32 
338083840*a^6*b^8*c^7*d*e^2 + 24763170816*a^7*b^5*c^9*d^2*e + 89087016960* 
a^7*b^6*c^8*d*e^2 + 31406948352*a^8*b^3*c^10*d^2*e - 117172076544*a^8*b^4* 
c^9*d*e^2 + 27380416512*a^9*b^2*c^10*d*e^2)/(16384*(b^14 - 16384*a^7*c^7 + 
 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^ 
5 + 28672*a^6*b^2*c^6 - 28*a*b^12*c)) - (x*(-(a*b^19*e^4 + 81*b^17*c^3*d^4 
 - 1184*a*b^15*c^4*d^4 - 983040*a^8*b*c^11*d^4 + a*b^4*e^4*(-(4*a*c - b^2) 
^15)^(1/2) + 4*a*c^4*d^4*(-(4*a*c - b^2)^15)^(1/2) - 3*a^2*b^17*c*e^4 + 12 
386304*a^10*b*c^9*e^4 + 1572864*a^9*c^11*d^3*e - 14155776*a^10*c^10*d*e^3 
+ 960*a^2*b^13*c^5*d^4 + 84480*a^3*b^11*c^6*d^4 - 719360*a^4*b^9*c^7*d^...
 

Reduce [F]

\[ \int \frac {x^6 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {x^{6} \left (e \,x^{4}+d \right )}{\left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:

int(x^6*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
 

Output:

int(x^6*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)