\(\int \frac {x (d+e x^4)}{(a+b x^4+c x^8)^2} \, dx\) [97]

Optimal result

Integrand size = 23, antiderivative size = 299 \[ \int \frac {x \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {x^2 \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+\frac {\sqrt {c} \left (b d-2 a e+\frac {b^2 d-12 a c d+4 a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b d-2 a e-\frac {b^2 d-12 a c d+4 a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

1/4*x^2*(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)+1/8*c^(1/2)*(b*d-2*a*e+(4*a*b*e-12*a*c*d+b^2*d)/(-4*a*c+b^2)^(1/2))* 
arctan(2^(1/2)*c^(1/2)*x^2/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a/(-4*a*c 
+b^2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/8*c^(1/2)*(b*d-2*a*e-(4*a*b*e-12*a*c* 
d+b^2*d)/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x^2/(b+(-4*a*c+b^2)^(1 
/2))^(1/2))*2^(1/2)/a/(-4*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.06 \[ \int \frac {x \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {\frac {2 x^2 \left (b^2 d+b \left (-a e+c d x^4\right )-2 a c \left (d+e x^4\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+\frac {\sqrt {2} \sqrt {c} \left (b^2 d+b \left (\sqrt {b^2-4 a c} d+4 a e\right )-2 a \left (6 c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b^2 d+12 a c d+b \sqrt {b^2-4 a c} d-4 a b e-2 a \sqrt {b^2-4 a c} e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{8 a} \] Input:

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

Output:

((2*x^2*(b^2*d + b*(-(a*e) + c*d*x^4) - 2*a*c*(d + e*x^4)))/((b^2 - 4*a*c) 
*(a + b*x^4 + c*x^8)) + (Sqrt[2]*Sqrt[c]*(b^2*d + b*(Sqrt[b^2 - 4*a*c]*d + 
 4*a*e) - 2*a*(6*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/ 
Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a 
*c]]) + (Sqrt[2]*Sqrt[c]*(-(b^2*d) + 12*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - 4* 
a*b*e - 2*a*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqr 
t[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(8*a)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1814, 1492, 25, 1480, 218}

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.

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

\(\Big \downarrow \) 1814

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

\(\Big \downarrow \) 1492

\(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}-\frac {\int -\frac {c (b d-2 a e) x^4+b^2 d-6 a c d+a b e}{c x^8+b x^4+a}dx^2}{2 a \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {c (b d-2 a e) x^4+b^2 d-6 a c d+a b e}{c x^8+b x^4+a}dx^2}{2 a \left (b^2-4 a c\right )}+\frac {x^2 \left (c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\right )\)

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

((x^2*(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^4))/(2*a*(b^2 - 4*a*c)* 
(a + b*x^4 + c*x^8)) + ((Sqrt[c]*(b*d - 2*a*e + (b^2*d - 12*a*c*d + 4*a*b* 
e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b - Sqrt[b^2 - 4*a 
*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(b*d - 2*a*e - (b^ 
2*d - 12*a*c*d + 4*a*b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/ 
Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a* 
(b^2 - 4*a*c)))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), 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^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

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]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.58

method	result	size
default	\(8 c^{2} \left (\frac {\frac {\left (-b d -\sqrt {-4 a c +b^{2}}\, d +2 a e \right ) \sqrt {-4 a c +b^{2}}\, x^{2}}{16 a c \left (x^{4}+\frac {b}{2 c}-\frac {\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {\left (12 \sqrt {-4 a c +b^{2}}\, a c d -3 \sqrt {-4 a c +b^{2}}\, b^{2} d -8 a^{2} c e -6 a \,b^{2} e +28 a b c d -3 b^{3} d \right ) \left (-2 b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a \left (4 a c +3 b^{2}\right ) \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{4 \sqrt {-4 a c +b^{2}}\, c \left (4 a c -b^{2}\right )}+\frac {\frac {\left (\sqrt {-4 a c +b^{2}}\, d +2 a e -b d \right ) \sqrt {-4 a c +b^{2}}\, x^{2}}{16 a c \left (x^{4}+\frac {\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}\right )}+\frac {\left (8 a^{2} c e +6 a \,b^{2} e -28 a b c d +3 b^{3} d +12 \sqrt {-4 a c +b^{2}}\, a c d -3 \sqrt {-4 a c +b^{2}}\, b^{2} d \right ) \left (2 b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a \left (4 a c +3 b^{2}\right ) \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{4 \sqrt {-4 a c +b^{2}}\, c \left (4 a c -b^{2}\right )}\right )\)	\(472\)
risch	\(\frac {\frac {c \left (2 a e -b d \right ) x^{6}}{4 a \left (4 a c -b^{2}\right )}+\frac {\left (a b e +2 a c d -d \,b^{2}\right ) x^{2}}{4 \left (4 a c -b^{2}\right ) a}}{c \,x^{8}+b \,x^{4}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4096 a^{9} c^{6}-6144 a^{8} b^{2} c^{5}+3840 a^{7} b^{4} c^{4}-1280 a^{6} b^{6} c^{3}+240 a^{5} b^{8} c^{2}-24 a^{4} b^{10} c +a^{3} b^{12}\right ) \textit {\_Z}^{4}+\left (-768 a^{6} b \,c^{4} e^{2}+3072 a^{6} c^{5} d e +512 a^{5} b^{3} c^{3} e^{2}-1536 a^{5} b^{2} c^{4} d e -3840 a^{5} b \,c^{5} d^{2}-96 a^{4} b^{5} c^{2} e^{2}-128 a^{4} b^{4} c^{3} d e +3840 a^{4} b^{3} c^{4} d^{2}+192 a^{3} b^{6} c^{2} d e -1504 a^{3} b^{5} c^{3} d^{2}+a^{2} b^{9} e^{2}-36 a^{2} b^{8} c d e +288 a^{2} b^{7} c^{2} d^{2}+2 a \,b^{10} d e -27 b^{9} a c \,d^{2}+b^{11} d^{2}\right ) \textit {\_Z}^{2}+16 a^{4} c^{3} e^{4}+24 a^{3} b^{2} c^{2} e^{4}-224 a^{3} b \,c^{3} d \,e^{3}+288 a^{3} c^{4} d^{2} e^{2}+9 a^{2} b^{4} c \,e^{4}-144 a^{2} b^{3} c^{2} d \,e^{3}+960 a^{2} b^{2} c^{3} d^{2} e^{2}-2016 a^{2} b \,c^{4} d^{3} e +1296 a^{2} c^{5} d^{4}+18 a \,b^{5} c d \,e^{3}-198 a \,b^{4} c^{2} d^{2} e^{2}+496 a \,b^{3} c^{3} d^{3} e -360 a \,b^{2} c^{4} d^{4}+9 b^{6} c \,d^{2} e^{2}-30 c^{2} b^{5} d^{3} e +25 b^{4} c^{3} d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-128 a^{5} c^{4} e +32 a^{4} b^{2} c^{3} e +448 a^{4} b \,c^{4} d +24 a^{3} b^{4} c^{2} e -400 a^{3} b^{3} c^{3} d -10 a^{2} b^{6} c e +132 a^{2} b^{5} c^{2} d +a \,b^{8} e -19 a \,b^{7} c d +b^{9} d \right ) \textit {\_R}^{2}+12 a^{2} b \,c^{2} e^{3}-24 a^{2} c^{3} d \,e^{2}+9 a \,b^{3} c \,e^{3}-102 a \,b^{2} c^{2} d \,e^{2}+276 a b \,c^{3} d^{2} e -216 a \,c^{4} d^{3}+9 b^{4} c d \,e^{2}-33 b^{3} c^{2} d^{2} e +30 b^{2} c^{3} d^{3}\right ) x^{2}+\left (256 a^{6} b \,c^{4}-256 a^{5} b^{3} c^{3}+96 a^{4} b^{5} c^{2}-16 a^{3} b^{7} c +a^{2} b^{9}\right ) \textit {\_R}^{3}+\left (32 a^{4} c^{3} e^{2}-48 a^{3} b^{2} c^{2} e^{2}+160 a^{3} b \,c^{3} d e -288 a^{3} c^{4} d^{2}+10 a^{2} b^{4} c \,e^{2}-48 a^{2} b^{3} c^{2} d e +128 a^{2} b^{2} c^{3} d^{2}+2 a \,b^{5} c d e -18 a \,b^{4} c^{2} d^{2}+b^{6} c \,d^{2}\right ) \textit {\_R} \right )\right )}{8}\)	\(944\)

Input:

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

Output:

8*c^2*(1/4/(-4*a*c+b^2)^(1/2)/c/(4*a*c-b^2)*(1/16*(-b*d-(-4*a*c+b^2)^(1/2) 
*d+2*a*e)*(-4*a*c+b^2)^(1/2)/a/c*x^2/(x^4+1/2*b/c-1/2*(-4*a*c+b^2)^(1/2)/c 
)+1/16*(12*(-4*a*c+b^2)^(1/2)*a*c*d-3*(-4*a*c+b^2)^(1/2)*b^2*d-8*a^2*c*e-6 
*a*b^2*e+28*a*b*c*d-3*b^3*d)*(-2*b+(-4*a*c+b^2)^(1/2))/a/(4*a*c+3*b^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)))+1/4/(-4*a*c+b^2)^(1/2)/c/(4*a*c-b^2)*(1/16*((-4*a*c 
+b^2)^(1/2)*d+2*a*e-b*d)*(-4*a*c+b^2)^(1/2)/a/c*x^2/(x^4+1/2*(-4*a*c+b^2)^ 
(1/2)/c+1/2*b/c)+1/16*(8*a^2*c*e+6*a*b^2*e-28*a*b*c*d+3*b^3*d+12*(-4*a*c+b 
^2)^(1/2)*a*c*d-3*(-4*a*c+b^2)^(1/2)*b^2*d)*(2*b+(-4*a*c+b^2)^(1/2))/a/(4* 
a*c+3*b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^2*2^(1/2)/( 
(b+(-4*a*c+b^2)^(1/2))*c)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4583 vs. \(2 (257) = 514\).

Time = 2.71 (sec) , antiderivative size = 4583, normalized size of antiderivative = 15.33 \[ \int \frac {x \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4435 vs. \(2 (257) = 514\).

Time = 5.62 (sec) , antiderivative size = 4435, normalized size of antiderivative = 14.83 \[ \int \frac {x \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(b*c*d*x^6 - 2*a*c*e*x^6 + b^2*d*x^2 - 2*a*c*d*x^2 - a*b*e*x^2)/((c*x^ 
8 + b*x^4 + a)*(a*b^2 - 4*a^2*c)) + 1/32*((2*b^3*c^2 - 8*a*b*c^3 - sqrt(2) 
*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 + 4*sqrt(2)*sqrt(b^ 
2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c + 2*sqrt(2)*sqrt(b^2 - 4* 
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqr 
t(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^2 - 2*(b^2 - 4*a*c)*b*c^2)*(a*b^2 - 4*a^2 
*c)^2*d - 2*(2*a*b^2*c^2 - 8*a^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
+ sqrt(b^2 - 4*a*c)*c)*a*b^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt 
(b^2 - 4*a*c)*c)*a^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 
 4*a*c)*c)*a*b*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)* 
c)*a*c^2 - 2*(b^2 - 4*a*c)*a*c^2)*(a*b^2 - 4*a^2*c)^2*e + 2*(sqrt(2)*sqrt( 
b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6 - 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c) 
*c)*a^2*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c - 2*a*b^ 
6*c + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^2 + 20*sqrt(2)* 
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 
- 4*a*c)*c)*a*b^4*c^2 + 28*a^2*b^4*c^2 - 96*sqrt(2)*sqrt(b*c + sqrt(b^2 - 
4*a*c)*c)*a^4*c^3 - 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^3 - 
 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 128*a^3*b^2*c^3 
+ 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 192*a^4*c^4 + 2*(b^ 
2 - 4*a*c)*a*b^4*c - 20*(b^2 - 4*a*c)*a^2*b^2*c^2 + 48*(b^2 - 4*a*c)*a^...
 

Mupad [B] (verification not implemented)

Time = 29.82 (sec) , antiderivative size = 32336, normalized size of antiderivative = 108.15 \[ \int \frac {x \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^2*(a*b*e - b^2*d + 2*a*c*d))/(4*a*(4*a*c - b^2)) + (c*x^6*(2*a*e - b*d 
))/(4*a*(4*a*c - b^2)))/(a + b*x^4 + c*x^8) - atan(((((((65536*a^8*b*c^9*e 
^2 - 589824*a^7*b*c^10*d^2 + 128*a^2*b^11*c^5*d^2 - 3328*a^3*b^9*c^6*d^2 + 
 36864*a^4*b^7*c^7*d^2 - 204800*a^5*b^5*c^8*d^2 + 557056*a^6*b^3*c^9*d^2 + 
 1280*a^4*b^9*c^5*e^2 - 16384*a^5*b^7*c^6*e^2 + 73728*a^6*b^5*c^7*e^2 - 13 
1072*a^7*b^3*c^8*e^2 + 256*a^3*b^10*c^5*d*e - 8192*a^4*b^8*c^6*d*e + 73728 
*a^5*b^6*c^7*d*e - 262144*a^6*b^4*c^8*d*e + 327680*a^7*b^2*c^9*d*e)/(a^3*b 
^8 + 256*a^7*c^4 - 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3) + ((x^ 
2*(4096*a^2*b^16*c^4*d - 126976*a^3*b^14*c^5*d + 1671168*a^4*b^12*c^6*d - 
12124160*a^5*b^10*c^7*d + 52428800*a^6*b^8*c^8*d - 135266304*a^7*b^6*c^9*d 
 + 192937984*a^8*b^4*c^10*d - 117440512*a^9*b^2*c^11*d + 4096*a^3*b^15*c^4 
*e - 90112*a^4*b^13*c^5*e + 786432*a^5*b^11*c^6*e - 3276800*a^6*b^9*c^7*e 
+ 5242880*a^7*b^7*c^8*e + 6291456*a^8*b^5*c^9*e - 33554432*a^9*b^3*c^10*e 
+ 33554432*a^10*b*c^11*e))/(8*(a^3*b^10 - 1024*a^8*c^5 - 20*a^4*b^8*c + 16 
0*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4)) + ((-(b^11*d^2 + a^2* 
b^9*e^2 + a^2*e^2*(-(4*a*c - b^2)^9)^(1/2) + b^2*d^2*(-(4*a*c - b^2)^9)^(1 
/2) - 3840*a^5*b*c^5*d^2 - 768*a^6*b*c^4*e^2 + 2*a*b^10*d*e + 288*a^2*b^7* 
c^2*d^2 - 1504*a^3*b^5*c^3*d^2 + 3840*a^4*b^3*c^4*d^2 - 96*a^4*b^5*c^2*e^2 
 + 512*a^5*b^3*c^3*e^2 - 27*a*b^9*c*d^2 - 9*a*c*d^2*(-(4*a*c - b^2)^9)^(1/ 
2) + 3072*a^6*c^5*d*e - 36*a^2*b^8*c*d*e + 192*a^3*b^6*c^2*d*e - 128*a^...
 

Reduce [F]

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

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

Output:

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