\(\int \frac {d+e x^4}{x^5 (a+b x^4+c x^8)^2} \, dx\) [93]

Optimal result

Integrand size = 25, antiderivative size = 225 \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )^2} \, dx=\frac {-2 b^2 d+6 a c d+a b e}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x^4}{4 a \left (b^2-4 a c\right ) x^4 \left (a+b x^4+c x^8\right )}-\frac {\left (2 b^4 d-12 a b^2 c d+12 a^2 c^2 d-a b^3 e+6 a^2 b c e\right ) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {(2 b d-a e) \log (x)}{a^3}+\frac {(2 b d-a e) \log \left (a+b x^4+c x^8\right )}{8 a^3} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.26 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.31 \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )^2} \, dx=\frac {-\frac {a d}{x^4}-\frac {a \left (b^3 d+2 a c \left (a e-c d x^4\right )+b^2 \left (-a e+c d x^4\right )-a b c \left (3 d+e x^4\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+4 (-2 b d+a e) \log (x)+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {2 b^4 d \log (x-\text {$\#$1})-10 a b^2 c d \log (x-\text {$\#$1})+6 a^2 c^2 d \log (x-\text {$\#$1})-a b^3 e \log (x-\text {$\#$1})+5 a^2 b c e \log (x-\text {$\#$1})+2 b^3 c d \log (x-\text {$\#$1}) \text {$\#$1}^4-8 a b c^2 d \log (x-\text {$\#$1}) \text {$\#$1}^4-a b^2 c e \log (x-\text {$\#$1}) \text {$\#$1}^4+4 a^2 c^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b+2 c \text {$\#$1}^4}\&\right ]}{b^2-4 a c}}{4 a^3} \] Input:

Integrate[(d + e*x^4)/(x^5*(a + b*x^4 + c*x^8)^2),x]
 

Output:

(-((a*d)/x^4) - (a*(b^3*d + 2*a*c*(a*e - c*d*x^4) + b^2*(-(a*e) + c*d*x^4) 
 - a*b*c*(3*d + e*x^4)))/((b^2 - 4*a*c)*(a + b*x^4 + c*x^8)) + 4*(-2*b*d + 
 a*e)*Log[x] + RootSum[a + b*#1^4 + c*#1^8 & , (2*b^4*d*Log[x - #1] - 10*a 
*b^2*c*d*Log[x - #1] + 6*a^2*c^2*d*Log[x - #1] - a*b^3*e*Log[x - #1] + 5*a 
^2*b*c*e*Log[x - #1] + 2*b^3*c*d*Log[x - #1]*#1^4 - 8*a*b*c^2*d*Log[x - #1 
]*#1^4 - a*b^2*c*e*Log[x - #1]*#1^4 + 4*a^2*c^2*e*Log[x - #1]*#1^4)/(b + 2 
*c*#1^4) & ]/(b^2 - 4*a*c))/(4*a^3)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1802, 1235, 25, 1200, 2009}

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.

\(\displaystyle \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )^2} \, dx\)

\(\Big \downarrow \) 1802

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1200

\(\displaystyle \frac {1}{4} \left (\frac {\int \left (-\frac {\left (4 a c-b^2\right ) (a e-2 b d)}{a^2 x^4}+\frac {2 d b^4-a e b^3-10 a c d b^2+5 a^2 c e b+c \left (b^2-4 a c\right ) (2 b d-a e) x^4+6 a^2 c^2 d}{a^2 \left (c x^8+b x^4+a\right )}+\frac {2 d b^2-a e b-6 a c d}{a x^8}\right )dx^4}{a \left (b^2-4 a c\right )}+\frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{a x^4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \left (\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right ) \left (6 a^2 b c e+12 a^2 c^2 d-a b^3 e-12 a b^2 c d+2 b^4 d\right )}{a^2 \sqrt {b^2-4 a c}}-\frac {\log \left (x^4\right ) \left (b^2-4 a c\right ) (2 b d-a e)}{a^2}+\frac {\left (b^2-4 a c\right ) (2 b d-a e) \log \left (a+b x^4+c x^8\right )}{2 a^2}-\frac {-a b e-6 a c d+2 b^2 d}{a x^4}}{a \left (b^2-4 a c\right )}+\frac {c x^4 (b d-2 a e)-a b e-2 a c d+b^2 d}{a x^4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\right )\)

Input:

Int[(d + e*x^4)/(x^5*(a + b*x^4 + c*x^8)^2),x]
 

Output:

((b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^4)/(a*(b^2 - 4*a*c)*x^4*(a + 
 b*x^4 + c*x^8)) + (-((2*b^2*d - 6*a*c*d - a*b*e)/(a*x^4)) - ((2*b^4*d - 1 
2*a*b^2*c*d + 12*a^2*c^2*d - a*b^3*e + 6*a^2*b*c*e)*ArcTanh[(b + 2*c*x^4)/ 
Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)*(2*b*d - a*e) 
*Log[x^4])/a^2 + ((b^2 - 4*a*c)*(2*b*d - a*e)*Log[a + b*x^4 + c*x^8])/(2*a 
^2))/(a*(b^2 - 4*a*c)))/4
 

Defintions of rubi rules used

rule 25

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

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 1802

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( 
e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1 
)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, 
c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.35

method	result	size
default	\(-\frac {\frac {\frac {a c \left (a b e +2 a c d -d \,b^{2}\right ) x^{4}}{4 a c -b^{2}}-\frac {a \left (2 a^{2} c e -a \,b^{2} e -3 a b c d +b^{3} d \right )}{4 a c -b^{2}}}{2 c \,x^{8}+2 b \,x^{4}+2 a}+\frac {\frac {\left (4 a^{2} c^{2} e -a \,b^{2} c e -8 d \,c^{2} b a +2 d c \,b^{3}\right ) \ln \left (c \,x^{8}+b \,x^{4}+a \right )}{2 c}+\frac {2 \left (5 a^{2} b c e +6 a^{2} c^{2} d -a \,b^{3} e -10 a \,b^{2} c d +2 b^{4} d -\frac {\left (4 a^{2} c^{2} e -a \,b^{2} c e -8 d \,c^{2} b a +2 d c \,b^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{8 a c -2 b^{2}}}{2 a^{3}}-\frac {d}{4 a^{2} x^{4}}+\frac {\left (a e -2 b d \right ) \ln \left (x \right )}{a^{3}}\)	\(303\)
risch	\(\frac {-\frac {c \left (a b e +6 a c d -2 d \,b^{2}\right ) x^{8}}{4 a^{2} \left (4 a c -b^{2}\right )}+\frac {\left (2 a^{2} c e -a \,b^{2} e -7 a b c d +2 b^{3} d \right ) x^{4}}{4 \left (4 a c -b^{2}\right ) a^{2}}-\frac {d}{4 a}}{x^{4} \left (c \,x^{8}+b \,x^{4}+a \right )}+\frac {\ln \left (x \right ) e}{a^{2}}-\frac {2 \ln \left (x \right ) b d}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{6} c^{3}-48 a^{5} b^{2} c^{2}+12 a^{4} b^{4} c -a^{3} b^{6}\right ) \textit {\_Z}^{2}+\left (64 a^{4} c^{3} e -48 a^{3} b^{2} c^{2} e -128 a^{3} b \,c^{3} d +12 a^{2} b^{4} c e +96 a^{2} b^{3} c^{2} d -a \,b^{6} e -24 a \,b^{5} c d +2 b^{7} d \right ) \textit {\_Z} +16 a^{2} c^{3} e^{2}-3 a \,b^{2} c^{2} e^{2}-28 a b \,c^{3} d e +36 a \,c^{4} d^{2}+6 b^{3} c^{2} d e -8 b^{2} c^{3} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (288 a^{7} c^{3}-224 a^{6} b^{2} c^{2}+58 a^{5} b^{4} c -5 a^{4} b^{6}\right ) \textit {\_R}^{2}+\left (144 a^{5} c^{3} e -68 a^{4} b^{2} c^{2} e -264 a^{4} b \,c^{3} d +8 a^{3} b^{4} c e +130 a^{3} b^{3} c^{2} d -16 a^{2} b^{5} c d \right ) \textit {\_R} +4 a^{2} b^{2} c^{2} e^{2}+48 a^{2} b \,c^{3} d e +144 a^{2} c^{4} d^{2}-16 a \,b^{3} c^{2} d e -96 a \,b^{2} c^{3} d^{2}+16 b^{4} c^{2} d^{2}\right ) x^{4}+\left (-16 a^{7} b \,c^{2}+8 a^{6} b^{3} c -a^{5} b^{5}\right ) \textit {\_R}^{2}+\left (68 a^{5} b \,c^{2} e +24 a^{5} c^{3} d -33 a^{4} b^{3} c e -142 a^{4} b^{2} c^{2} d +4 a^{3} b^{5} e +66 a^{3} b^{4} c d -8 a^{2} b^{6} d \right ) \textit {\_R} -16 a^{3} b \,c^{2} e^{2}-96 a^{3} c^{3} d e +4 a^{2} b^{3} c \,e^{2}+88 a^{2} b^{2} c^{2} d e +192 a^{2} b \,c^{3} d^{2}-16 a \,b^{4} c d e -112 a \,b^{3} c^{2} d^{2}+16 b^{5} c \,d^{2}\right )\right )}{4}\)	\(692\)

Input:

int((e*x^4+d)/x^5/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/2/a^3*(1/2*(a*c*(a*b*e+2*a*c*d-b^2*d)/(4*a*c-b^2)*x^4-a*(2*a^2*c*e-a*b^ 
2*e-3*a*b*c*d+b^3*d)/(4*a*c-b^2))/(c*x^8+b*x^4+a)+1/2/(4*a*c-b^2)*(1/2*(4* 
a^2*c^2*e-a*b^2*c*e-8*a*b*c^2*d+2*b^3*c*d)/c*ln(c*x^8+b*x^4+a)+2*(5*a^2*b* 
c*e+6*a^2*c^2*d-a*b^3*e-10*a*b^2*c*d+2*b^4*d-1/2*(4*a^2*c^2*e-a*b^2*c*e-8* 
a*b*c^2*d+2*b^3*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2) 
^(1/2))))-1/4*d/a^2/x^4+(a*e-2*b*d)/a^3*ln(x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 812 vs. \(2 (214) = 428\).

Time = 19.23 (sec) , antiderivative size = 1647, normalized size of antiderivative = 7.32 \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x^4+d)/x^5/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
 

Output:

[-1/8*(2*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d - (a^2*b^3*c - 4*a^3* 
b*c^2)*e)*x^8 + 2*((2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2)*d - (a^2*b^4 - 
6*a^3*b^2*c + 8*a^4*c^2)*e)*x^4 + ((2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d 
- (a*b^3*c - 6*a^2*b*c^2)*e)*x^12 + (2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d - 
 (a*b^4 - 6*a^2*b^2*c)*e)*x^8 + (2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d - ( 
a^2*b^3 - 6*a^3*b*c)*e)*x^4)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^8 + 2*b*c*x^4 
+ b^2 - 2*a*c + (2*c*x^4 + b)*sqrt(b^2 - 4*a*c))/(c*x^8 + b*x^4 + a)) + 2* 
(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d - ((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2 
*b*c^3)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^12 + (2*(b^6 - 8*a 
*b^4*c + 16*a^2*b^2*c^2)*d - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^8 + 
 (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16*a 
^4*c^2)*e)*x^4)*log(c*x^8 + b*x^4 + a) + 8*((2*(b^5*c - 8*a*b^3*c^2 + 16*a 
^2*b*c^3)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^12 + (2*(b^6 - 8 
*a*b^4*c + 16*a^2*b^2*c^2)*d - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^8 
 + (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16 
*a^4*c^2)*e)*x^4)*log(x))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^12 + 
 (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^8 + (a^4*b^4 - 8*a^5*b^2*c + 16* 
a^6*c^2)*x^4), -1/8*(2*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d - (a^2* 
b^3*c - 4*a^3*b*c^2)*e)*x^8 + 2*((2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2)*d 
 - (a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*e)*x^4 + 2*((2*(b^4*c - 6*a*b^2*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate((e*x**4+d)/x**5/(c*x**8+b*x**4+a)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x^4+d)/x^5/(c*x^8+b*x^4+a)^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 4.80 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.13 \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )^2} \, dx=\frac {{\left (2 \, b^{4} d - 12 \, a b^{2} c d + 12 \, a^{2} c^{2} d - a b^{3} e + 6 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{2} c d x^{8} - 6 \, a c^{2} d x^{8} - a b c e x^{8} + 2 \, b^{3} d x^{4} - 7 \, a b c d x^{4} - a b^{2} e x^{4} + 2 \, a^{2} c e x^{4} + a b^{2} d - 4 \, a^{2} c d}{4 \, {\left (c x^{12} + b x^{8} + a x^{4}\right )} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} + \frac {{\left (2 \, b d - a e\right )} \log \left (c x^{8} + b x^{4} + a\right )}{8 \, a^{3}} - \frac {{\left (2 \, b d - a e\right )} \log \left (x^{4}\right )}{4 \, a^{3}} \] Input:

integrate((e*x^4+d)/x^5/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
 

Output:

1/4*(2*b^4*d - 12*a*b^2*c*d + 12*a^2*c^2*d - a*b^3*e + 6*a^2*b*c*e)*arctan 
((2*c*x^4 + b)/sqrt(-b^2 + 4*a*c))/((a^3*b^2 - 4*a^4*c)*sqrt(-b^2 + 4*a*c) 
) - 1/4*(2*b^2*c*d*x^8 - 6*a*c^2*d*x^8 - a*b*c*e*x^8 + 2*b^3*d*x^4 - 7*a*b 
*c*d*x^4 - a*b^2*e*x^4 + 2*a^2*c*e*x^4 + a*b^2*d - 4*a^2*c*d)/((c*x^12 + b 
*x^8 + a*x^4)*(a^2*b^2 - 4*a^3*c)) + 1/8*(2*b*d - a*e)*log(c*x^8 + b*x^4 + 
 a)/a^3 - 1/4*(2*b*d - a*e)*log(x^4)/a^3
 

Mupad [B] (verification not implemented)

Time = 108.62 (sec) , antiderivative size = 47178, normalized size of antiderivative = 209.68 \[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

int((d + e*x^4)/(x^5*(a + b*x^4 + c*x^8)^2),x)
 

Output:

(log(x)*(a*e - 2*b*d))/a^3 - (d/(4*a) - (x^4*(2*b^3*d - a*b^2*e + 2*a^2*c* 
e - 7*a*b*c*d))/(4*a^2*(4*a*c - b^2)) + (c*x^8*(a*b*e - 2*b^2*d + 6*a*c*d) 
)/(4*a^2*(4*a*c - b^2)))/(a*x^4 + b*x^8 + c*x^12) + (log((((a*e - 2*b*d + 
a^3*(-(2*b^4*d + 12*a^2*c^2*d - a*b^3*e - 12*a*b^2*c*d + 6*a^2*b*c*e)^2/(a 
^6*(4*a*c - b^2)^3))^(1/2))*(((a*e - 2*b*d + a^3*(-(2*b^4*d + 12*a^2*c^2*d 
 - a*b^3*e - 12*a*b^2*c*d + 6*a^2*b*c*e)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*( 
((a*e - 2*b*d + a^3*(-(2*b^4*d + 12*a^2*c^2*d - a*b^3*e - 12*a*b^2*c*d + 6 
*a^2*b*c*e)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*((((256*b^3*c^4*(2*b^4*d + 6*a 
^2*c^2*d - a*b^3*e - 10*a*b^2*c*d + 5*a^2*b*c*e))/(a^2*(4*a*c - b^2)) + (3 
2*b^3*c^4*(a*e - 2*b*d + a^3*(-(2*b^4*d + 12*a^2*c^2*d - a*b^3*e - 12*a*b^ 
2*c*d + 6*a^2*b*c*e)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*(a*b + 5*b^2*x^4 - 18 
*a*c*x^4))/a^3 - (64*b^2*c^5*x^4*(14*b^4*d + 324*a^2*c^2*d - 7*a*b^3*e - 1 
32*a*b^2*c*d + 18*a^2*b*c*e))/(a^2*(4*a*c - b^2)))*(a*e - 2*b*d + a^3*(-(2 
*b^4*d + 12*a^2*c^2*d - a*b^3*e - 12*a*b^2*c*d + 6*a^2*b*c*e)^2/(a^6*(4*a* 
c - b^2)^3))^(1/2)))/(8*a^3) + (32*b^2*c^5*(108*a^3*c^3*d^2 - 8*a^2*b^4*e^ 
2 - 32*b^6*d^2 + 35*a^3*b^2*c*e^2 + 32*a*b^5*d*e - 456*a^2*b^2*c^2*d^2 + 2 
36*a*b^4*c*d^2 - 188*a^2*b^3*c*d*e + 228*a^3*b*c^2*d*e))/(a^4*(4*a*c - b^2 
)^2) - (16*b*c^6*x^4*(52*b^6*d^2 + 13*a^2*b^4*e^2 + 1944*a^3*c^3*d^2 - 54* 
a^3*b^2*c*e^2 - 52*a*b^5*d*e - 504*a^2*b^2*c^2*d^2 - 204*a*b^4*c*d^2 + 210 
*a^2*b^3*c*d*e))/(a^4*(4*a*c - b^2)^2)))/(8*a^3) + (16*b*c^6*(a*b*e - 2...
 

Reduce [F]

\[ \int \frac {d+e x^4}{x^5 \left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {e \,x^{4}+d}{x^{5} \left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:

int((e*x^4+d)/x^5/(c*x^8+b*x^4+a)^2,x)
 

Output:

int((e*x^4+d)/x^5/(c*x^8+b*x^4+a)^2,x)