\(\int \frac {x^2 (d+e x^4)}{(a+b x^4+c x^8)^2} \, dx\) [102]

Optimal result

Integrand size = 25, antiderivative size = 542 \[ \int \frac {x^2 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {x^3 \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+\frac {\sqrt [4]{c} \left (b d-2 a e-\frac {b^2 d-20 a c d+8 a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a \left (b^2-4 a c\right ) \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (b d-2 a e+\frac {b^2 d-20 a c d+8 a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (b d-2 a e-\frac {b^2 d-20 a c d+8 a b e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a \left (b^2-4 a c\right ) \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (b d-2 a e+\frac {b^2 d-20 a c d+8 a b e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8\ 2^{3/4} a \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \] Output:

1/4*x^3*(b^2*d-2*a*c*d-a*b*e+c*(-2*a*e+b*d)*x^4)/a/(-4*a*c+b^2)/(c*x^8+b*x 
^4+a)+1/16*c^(1/4)*(b*d-2*a*e-(8*a*b*e-20*a*c*d+b^2*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)/a/(-4*a*c 
+b^2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)+1/16*c^(1/4)*(b*d-2*a*e+(8*a*b*e-20*a* 
c*d+b^2*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)/a/(-4*a*c+b^2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)-1/16*c^( 
1/4)*(b*d-2*a*e-(8*a*b*e-20*a*c*d+b^2*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)/a/(-4*a*c+b^2)/(-b-(-4 
*a*c+b^2)^(1/2))^(1/4)-1/16*c^(1/4)*(b*d-2*a*e+(8*a*b*e-20*a*c*d+b^2*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)/a/(-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 = 0.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.34 \[ \int \frac {x^2 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {4 x^3 \left (-b^2 d+b \left (a e-c d x^4\right )+2 a c \left (d+e x^4\right )\right )-\left (a+b x^4+c x^8\right ) \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^2 d \log (x-\text {$\#$1})-10 a c d \log (x-\text {$\#$1})+3 a b e \log (x-\text {$\#$1})+b c d \log (x-\text {$\#$1}) \text {$\#$1}^4-2 a c e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{16 a \left (-b^2+4 a c\right ) \left (a+b x^4+c x^8\right )} \] Input:

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

Output:

(4*x^3*(-(b^2*d) + b*(a*e - c*d*x^4) + 2*a*c*(d + e*x^4)) - (a + b*x^4 + c 
*x^8)*RootSum[a + b*#1^4 + c*#1^8 & , (b^2*d*Log[x - #1] - 10*a*c*d*Log[x 
- #1] + 3*a*b*e*Log[x - #1] + b*c*d*Log[x - #1]*#1^4 - 2*a*c*e*Log[x - #1] 
*#1^4)/(b*#1 + 2*c*#1^5) & ])/(16*a*(-b^2 + 4*a*c)*(a + b*x^4 + c*x^8))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 439, 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 = {1824, 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.

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

\(\Big \downarrow \) 1824

\(\displaystyle \frac {x^3 \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}-\frac {\int -\frac {x^2 \left (c (b d-2 a e) x^4+b^2 d-10 a c d+3 a b e\right )}{c x^8+b x^4+a}dx}{4 a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {x^2 \left (c (b d-2 a e) x^4+b^2 d-10 a c d+3 a b e\right )}{c x^8+b x^4+a}dx}{4 a \left (b^2-4 a c\right )}+\frac {x^3 \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 1834

\(\displaystyle \frac {\frac {1}{2} c \left (\frac {8 a b e-20 a c d+b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \int \frac {2 x^2}{2 c x^4+b-\sqrt {b^2-4 a c}}dx+\frac {1}{2} c \left (-\frac {8 a b e-20 a c d+b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \int \frac {2 x^2}{2 c x^4+b+\sqrt {b^2-4 a c}}dx}{4 a \left (b^2-4 a c\right )}+\frac {x^3 \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c \left (\frac {8 a b e-20 a c d+b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \int \frac {x^2}{2 c x^4+b-\sqrt {b^2-4 a c}}dx+c \left (-\frac {8 a b e-20 a c d+b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \int \frac {x^2}{2 c x^4+b+\sqrt {b^2-4 a c}}dx}{4 a \left (b^2-4 a c\right )}+\frac {x^3 \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {c \left (-\frac {8 a b e-20 a c d+b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {-b-\sqrt {b^2-4 a c}}}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )+c \left (\frac {8 a b e-20 a c d+b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {\sqrt {b^2-4 a c}-b}}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {x^3 \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {c \left (-\frac {8 a b e-20 a c d+b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )+c \left (\frac {8 a b e-20 a c d+b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {x^3 \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {c \left (-\frac {8 a b e-20 a c d+b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )+c \left (\frac {8 a b e-20 a c d+b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {x^3 \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\)

Input:

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

Output:

(x^3*(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^4))/(4*a*(b^2 - 4*a*c)*( 
a + b*x^4 + c*x^8)) + (c*(b*d - 2*a*e - (b^2*d - 20*a*c*d + 8*a*b*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))) + c*(b*d - 2*a*e + (b^2*d - 20*a*c*d + 8*a*b*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))))/(4*a*(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 1824

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( 
c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(a + b*x^n + c*x^ 
(2*n))^(p + 1)*((d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^n)/(a*f*n*(p + 
 1)*(b^2 - 4*a*c))), x] + Simp[1/(a*n*(p + 1)*(b^2 - 4*a*c))   Int[(f*x)^m* 
(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m 
+ 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + n*(2*p + 3) + 1)*(b*d - 2*a*e) 
*x^n, x], 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] && LtQ[p, -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 = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.32

method	result	size
default	\(\frac {\frac {c \left (2 a e -b d \right ) x^{7}}{4 a \left (4 a c -b^{2}\right )}+\frac {\left (a b e +2 a c d -d \,b^{2}\right ) x^{3}}{4 \left (4 a c -b^{2}\right ) a}}{c \,x^{8}+b \,x^{4}+a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (-2 a e +b d \right ) \textit {\_R}^{6}+\left (3 a b e -10 a c d +d \,b^{2}\right ) \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{16 a \left (4 a c -b^{2}\right )}\)	\(172\)
risch	\(\frac {\frac {c \left (2 a e -b d \right ) x^{7}}{4 a \left (4 a c -b^{2}\right )}+\frac {\left (a b e +2 a c d -d \,b^{2}\right ) x^{3}}{4 \left (4 a c -b^{2}\right ) a}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\frac {c \left (2 a e -b d \right ) \textit {\_R}^{6}}{4 a c -b^{2}}-\frac {\left (3 a b e -10 a c d +d \,b^{2}\right ) \textit {\_R}^{2}}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{16 a}\)	\(186\)

Input:

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

Output:

(1/4*c*(2*a*e-b*d)/a/(4*a*c-b^2)*x^7+1/4*(a*b*e+2*a*c*d-b^2*d)/(4*a*c-b^2) 
/a*x^3)/(c*x^8+b*x^4+a)-1/16/a/(4*a*c-b^2)*sum((c*(-2*a*e+b*d)*_R^6+(3*a*b 
*e-10*a*c*d+b^2*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. \[ \int \frac {x^2 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/4*((b*c*d - 2*a*c*e)*x^7 - (a*b*e - (b^2 - 2*a*c)*d)*x^3)/((a*b^2*c - 4* 
a^2*c^2)*x^8 + (a*b^3 - 4*a^2*b*c)*x^4 + a^2*b^2 - 4*a^3*c) - 1/4*integrat 
e(-((b*c*d - 2*a*c*e)*x^6 + (3*a*b*e + (b^2 - 10*a*c)*d)*x^2)/(c*x^8 + b*x 
^4 + a), x)/(a*b^2 - 4*a^2*c)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 28.93 (sec) , antiderivative size = 93859, normalized size of antiderivative = 173.17 \[ \int \frac {x^2 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^3*(a*b*e - b^2*d + 2*a*c*d))/(4*a*(4*a*c - b^2)) + (c*x^7*(2*a*e - b*d 
))/(4*a*(4*a*c - b^2)))/(a + b*x^4 + c*x^8) - atan(((((65536*b^19*c^4*d^3 
- 2147483648*a^11*c^12*e^3 - 3735552*a*b^17*c^5*d^3 - 348966092800*a^9*b*c 
^13*d^3 + 161061273600*a^10*c^13*d^2*e + 91291648*a^2*b^15*c^6*d^3 - 12559 
31904*a^3*b^13*c^7*d^3 + 10742661120*a^4*b^11*c^8*d^3 - 59437481984*a^5*b^ 
9*c^9*d^3 + 213456519168*a^6*b^7*c^10*d^3 - 481371881472*a^7*b^5*c^11*d^3 
+ 620354338816*a^8*b^3*c^12*d^3 + 1769472*a^3*b^16*c^4*e^3 - 38928384*a^4* 
b^14*c^5*e^3 + 339214336*a^5*b^12*c^6*e^3 - 1402994688*a^6*b^10*c^7*e^3 + 
2139095040*a^7*b^8*c^8*e^3 + 3388997632*a^8*b^6*c^9*e^3 - 16508780544*a^9* 
b^4*c^10*e^3 + 17716740096*a^10*b^2*c^11*e^3 + 589824*a*b^18*c^4*d^2*e - 9 
3415538688*a^10*b*c^12*d*e^2 - 26738688*a^2*b^16*c^5*d^2*e + 1769472*a^2*b 
^17*c^4*d*e^2 + 506068992*a^3*b^14*c^6*d^2*e - 59572224*a^3*b^15*c^5*d*e^2 
 - 5236064256*a^4*b^12*c^7*d^2*e + 812384256*a^4*b^13*c^6*d*e^2 + 32432455 
680*a^5*b^10*c^8*d^2*e - 5822742528*a^5*b^11*c^7*d*e^2 - 122532397056*a^6* 
b^8*c^9*d^2*e + 23215472640*a^6*b^9*c^8*d*e^2 + 269475643392*a^7*b^6*c^10* 
d^2*e - 47362080768*a^7*b^7*c^9*d*e^2 - 284675801088*a^8*b^4*c^11*d^2*e + 
24763170816*a^8*b^5*c^10*d*e^2 + 14495514624*a^9*b^2*c^12*d^2*e + 70061654 
016*a^9*b^3*c^11*d*e^2)/(16384*(a^2*b^14 - 16384*a^9*c^7 - 28*a^3*b^12*c + 
 336*a^4*b^10*c^2 - 2240*a^5*b^8*c^3 + 8960*a^6*b^6*c^4 - 21504*a^7*b^4*c^ 
5 + 28672*a^8*b^2*c^6)) - (x*(-(b^21*d^4 + 81*a^4*b^17*e^4 + b^6*d^4*(-...
 

Reduce [F]

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

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

Output:

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